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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Fast-slow dynamical approximation of forced impact systems near periodic solutions

Flaviano Battelli1 and Michal Fečkan2*

Author Affiliations

1 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Marche Polytecnic University, Ancona, 60100, Italy

2 Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, Bratislava, 842 48, Slovakia

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Boundary Value Problems 2013, 2013:71  doi:10.1186/1687-2770-2013-71

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/71


Received:14 October 2012
Accepted:21 March 2013
Published:3 April 2013

© 2013 Battelli and Fečkan; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We approximate impact systems in arbitrary finite dimensions with fast-slow dynamics represented by regular ODE on one side of the impact manifold and singular ODE on the other. Lyapunov-Schmidt method leading to Poincaré-Melnikov function is applied to study bifurcations of periodic solutions. Several examples are presented as illustrations of abstract theory.

MSC: 34C23, 34C25, 37G15, 70K50.

Keywords:
fast-slow dynamics; impact systems; bifurcations; periodic solutions; Poincaré-Melnikov method

1 Introduction

Non-smooth differential equations when the vector field is only piecewise smooth, occur in various situations: in mechanical systems with dry frictions or with impacts, in control theory, electronics, economics, medicine and biology (see [1-6] for more references). One way of studying non-smooth systems is a regularization process consisting on approximation of the discontinuous vector field by a one-parametric family of smooth vector fields, which is called a regularization of the discontinuous one. The main problem then is to preserve certain dynamical properties of the original one to the regularized system. According to our knowledge, the regularization method has been mostly used to differential equations with non-smooth nonlinearities, like dry friction nonlinearity (see [7] and a survey paper [8]). As it is shown in [7,8], the regularization process is closely connected to a geometric singular perturbation theory [9,10]. On the other hand, it is argued in [11] that a harmonic oscillator with a jumping non-linearity with the force field nearly infinite in one side is a better model for describing the bouncing ball, rather then its limit version for an impact oscillator. This approach is used also in [12] when an impact oscillator is approximated by a one-parametric family of singularly perturbed differential equations, but as discussed in [12], the geometric singular perturbation theory does not apply.

In this paper, we continue in a spirit of [12] as follows. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M1">View MathML</a> be an open subset and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M2">View MathML</a> a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-function, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M4">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M5">View MathML</a>. Then S is a smooth hyper-surface of Ω that we call impact manifold, (or hyper-surface). We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M6">View MathML</a> and consider the following regular-singular perturbed system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M7">View MathML</a>

(1.1)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a> small. We assume that the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M9">View MathML</a>

(1.2)

has a continuous periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M10">View MathML</a> crossing transversally the impact manifold S, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M11">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M13">View MathML</a>. By transversal crossing, we mean that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M14">View MathML</a>

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M15">View MathML</a> and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M16">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17">View MathML</a>-periodic in t.

Transversal crossing implies that (1.2) has a family of continuous solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M18">View MathML</a>, α∈ (an open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M19">View MathML</a> of 0∈) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20">View MathML</a> crossing transversally the impact manifold S, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M21">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M23">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M25">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M26">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a> in α, and the maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M29">View MathML</a> give smooth (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>) parameterizations of the manifold S in small neighborhoods <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M31">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M33">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M34">View MathML</a>. Then the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M35">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M36">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-smooth. In this paper, we study the problem of existence of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17">View MathML</a>-periodic solution of the singular problem (1.1) in a neighborhood of the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M39">View MathML</a>

As a matter of fact, in the time interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M40">View MathML</a>, resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M41">View MathML</a>, the periodic solutions will stay close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M42">View MathML</a>, resp. to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43">View MathML</a>, and hence it will pass from the point of S near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M32">View MathML</a> to the point of S near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M45">View MathML</a> in a very short time (of the size of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M46">View MathML</a>). So, we may say that the behavior of the periodic solutions of (1.1) in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M47">View MathML</a> is quite well simulated by the solution of the perturbed impact system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M48">View MathML</a>

(1.3)

It is now clear that our study has been mostly motivated by the paper [12], where a similar problem on planar perturbed harmonic oscillators is studied. However arguments in [12] are mainly based on averaging methods whereas, in this paper, we investigate a general higher-dimensional singular equation such as (1.1) by using the Lyapunov-Schmidt reduction. We focus on the existence of periodic solutions and do not check their local asymptotic properties as, for example, stability or hyperbolicity. This could be also done by following our approach but we do not go into detail in this paper.

Our results (see Theorems 3.1 and 5.1) state that if a certain Poincaré-Melnikov-like function has a simple zero then the above problem has an affirmative answer. The proof of this fact is accomplished in several steps. In Section 2, we show, for any α in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, the existence of a unique continuous solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M50">View MathML</a> of (1.1) near the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M51">View MathML</a> which is defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M53">View MathML</a> and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M54">View MathML</a>, for some τ, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M56">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M57">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M59">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a> close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M61">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M59">View MathML</a> give <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a> parameterizations of S in neighborhoods of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M65">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M66">View MathML</a> gives a Poincaré-like map and a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M67">View MathML</a>-periodic solution is found by solving the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M68">View MathML</a>

Thus, the bifurcation equation is obtained by putting conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M70">View MathML</a> and the fact that the points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M72">View MathML</a> belong to S together. Then, in Section 3, we use the Lyapunov-Schmidt method to prove that the above equations can be solved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M73">View MathML</a> as functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a> small provided a certain Poincaré-Melnikov-like function has a simple zero. We will first study the case, that we call non-degenerate, when

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M75">View MathML</a>

(1.4)

Condition (1.4) has a simple geometrical meaning. The impact system (1.3) has a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M76">View MathML</a>-periodic solution if and only if the following condition holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M77">View MathML</a>

(1.5)

Now, suppose there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M78">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M79">View MathML</a> such that (1.5) holds. Possibly passing to a subsequence we can suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M80">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M81">View MathML</a>. Then, taking the limit in the equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M82">View MathML</a>

we see that condition (1.4) does not hold. Thus, (1.4) implies that, in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, there are no other <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M76">View MathML</a>-periodic solutions of (1.3) apart from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43">View MathML</a>.

In Section 4, we define the adjoint system to the linearization of the impact system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M86">View MathML</a>

(1.6)

along the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M87">View MathML</a> and relate the Poincaré-Melnikov function obtained in Section 3 with the solutions of such an adjoint system.

Section 5 is devoted to the extension of the result to the case (that we call degenerate) where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M88">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89">View MathML</a>. We will see that our results can be easily extended provided one of the following two conditions hold:

either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M91">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89">View MathML</a>.

Section 6 is devoted to the construction of some planar examples, although our results are given for an arbitrary finite dimension. Finally, the Appendix contains some technical proofs.

2 The bifurcation equation

We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M94">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M95">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M96">View MathML</a>

and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M97">View MathML</a> is a continuous periodic solution, of period <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M98">View MathML</a>, of the piecewise continuous singular system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M99">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M100">View MathML</a> extends to a solution of the following impact system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M101">View MathML</a>

that can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M102">View MathML</a>

Our purpose is to find a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17">View MathML</a>-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104">View MathML</a> of system (1.1), which is orbitally close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M105">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M106">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M107">View MathML</a> that is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M108">View MathML</a>

(2.1)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M109">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M110">View MathML</a>. Thus, we may say that, in some sense, the impact periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M111">View MathML</a> approximates the periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104">View MathML</a> of the singular perturbed equation (1.1).

To this end, we first set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M113">View MathML</a> in equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M114">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M115">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M116">View MathML</a>

(2.2)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M117">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M118">View MathML</a> describes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M119">View MathML</a>, we consider (2.2) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M120">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M121">View MathML</a> be the fundamental solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M122">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M123">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M124">View MathML</a> is the fundamental solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M125">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M123">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127">View MathML</a> be near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a>. By the variation of constants formula, the solution of (2.2) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M120">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M130">View MathML</a>

Thus, we conclude that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127">View MathML</a> near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a>equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M134">View MathML</a>has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M135">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M136">View MathML</a>if and only if the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M137">View MathML</a>given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M138">View MathML</a>

(2.3)

has a fixed point whose sup-norm in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M139">View MathML</a>is smaller thanρ. To show that (2.3) has a fixed point of norm less than ρ, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M141">View MathML</a> and note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M135">View MathML</a> is a fixed point of (2.3) of norm less than ρ, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M143">View MathML</a>, if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M144">View MathML</a> is a fixed point of norm less than ρ of the map:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M145">View MathML</a>

(2.4)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M146">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M147">View MathML</a>

and hence in the fixed-point equation (2.4), we may also take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M148">View MathML</a>. Then since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M149">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M150">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-map and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M152">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M153">View MathML</a>

the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M154">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-contraction on the Banach space of bounded continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M156">View MathML</a> whose sup-norm is less than or equal to ρ provided ρ is sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127">View MathML</a> is near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M159">View MathML</a> is small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M161">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M162">View MathML</a> be the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-solution of the fixed point (2.4). We emphasize the fact that ε may also be non-positive. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M164">View MathML</a> is a fixed point of (2.3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M165">View MathML</a>

(2.5)

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a> in all parameters and t.

Writing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M167">View MathML</a> in place of t in (2.4) and using (2.5) we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M168">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M169">View MathML</a>. We have, by definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M170">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M171">View MathML</a>

if and only if (recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M172">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M173">View MathML</a>

(2.7)

We remark that equation (2.7) has meaning also when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M174">View MathML</a> but its relevance for our problem is only when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a>.

As second step we consider the solution of the differential equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M176">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M177">View MathML</a>

which is close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M178">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M180">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M181">View MathML</a> be the fundamental solution of the linear system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M183">View MathML</a>. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M184">View MathML</a> we see that (for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M185">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M186">View MathML</a> satisfies the equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M187">View MathML</a>

(2.8)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M188">View MathML</a>

Again by the variation of constants formula we get the integral formula:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M189">View MathML</a>

which, as before, has a unique solution of norm less than a given, small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M190">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M191">View MathML</a>. At <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M192">View MathML</a> the solution of (2.8) takes the value:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M193">View MathML</a>

Now, we want to solve the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M194">View MathML</a>

that is [again using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M172">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M196">View MathML</a>]:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M197">View MathML</a>

(2.9)

Of course, when (2.9) holds, then (2.7) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M198">View MathML</a>

(2.10)

So, our task reduces to solve the system formed by equations (2.9), (2.10) together with the period equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M199">View MathML</a>

that is the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M200">View MathML</a> where:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M201">View MathML</a>

According to the smoothness properties of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M202">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M203">View MathML</a>, it results that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M204">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>.

3 Solving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206">View MathML</a>

In this section, we will give a criterion to solve equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M208">View MathML</a> in terms of ε for small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a>. We will use a Crandall-Rabinowitz type result (see also [[13], Theorem 4.1]) concerning the existence of a solution of a nonlinear equation having a manifold of fixed point at a certain value of a parameter.

Our result is as follows. Consider the linear system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M210">View MathML</a>

(3.1)

We will prove that if (1.4) holds, system (3.1) has a unique solution, up to a multiplicative constant, and the following result holds:

Theorem 3.1Assume condition (1.4) holds and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M211">View MathML</a>be the unique (up to a multiplicative constant) solution of (3.1). If the Poincaré-Melnikov function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M212">View MathML</a>

(3.2)

has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M213">View MathML</a>, then system (1.1) has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17">View MathML</a>-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104">View MathML</a>satisfying (2.1).

Proof To start with, we make few remarks on the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M216">View MathML</a>. First we note that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a> equation (2.8) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M218">View MathML</a>

which has the (unique) solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M219">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M220">View MathML</a>

Next, differentiating equation (2.8) with respect to ε we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M221">View MathML</a> satisfies the equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M222">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M223">View MathML</a>

Next, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M224">View MathML</a> by the definition and differentiating equation (2.6) with respect to ε at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M225">View MathML</a> and using the equalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M226">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M227">View MathML</a>

So, equation (2.9) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M229">View MathML</a> becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M230">View MathML</a>

which is satisfied for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>. Now we look at equation (2.10). Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M232">View MathML</a>, we see that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M234">View MathML</a> the equality is satisfied. As a consequence, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M235">View MathML</a>

(3.3)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M236">View MathML</a>. Next we look at derivatives of ℱ with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M238">View MathML</a>, α and ε at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M239">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M240">View MathML</a>

and similarly, using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M241">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M242">View MathML</a>

Next

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M243">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M244">View MathML</a>

So, the Jacobian matrix L of ℱ at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M239">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M246">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M247">View MathML</a> belongs to the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M248">View MathML</a> of L if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M249">View MathML</a>

(3.4)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M250">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M251">View MathML</a>

(3.5)

thus, on account of the transversality condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M252">View MathML</a>, (3.4) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M253">View MathML</a>

(3.6)

Next, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M254">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M255">View MathML</a>

(3.7)

then subtracting (3.5) from (3.7) and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M256">View MathML</a> we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M257">View MathML</a>

So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258">View MathML</a> satisfies (3.6), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M259">View MathML</a>

and then, on account of transversality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M260">View MathML</a>. Summarizing, we have seen that, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M261">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M262">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M265">View MathML</a>

(3.8)

On the other hand, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258">View MathML</a> satisfies (3.8), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M267">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M248">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M269">View MathML</a> if and only if system (3.8) has the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M270">View MathML</a> only. But (3.8) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M271">View MathML</a>

and hence (3.8) has the trivial solution if and only if the non-degenerateness condition (1.4) holds. We emphasize the fact that, assuming condition (1.4), equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272">View MathML</a> has the manifold of fixed points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M273">View MathML</a> and the linearization of ℱ at these points is Fredholm with index zero with the one-dimensional kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M274">View MathML</a>. Hence, there is a unique vector, up to a multiplicative constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M275">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M276">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M277">View MathML</a>

Writing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M278">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M279">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M280">View MathML</a> we see that ψ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M281">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282">View MathML</a> satisfy (3.1). This proves the claim before the statement of Theorem 3.1.

We recall that our purpose is to solve the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M206">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272">View MathML</a> has the one-dimensional manifold of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M273">View MathML</a> and its linearization along the points of this manifold is Fredholm with the one-dimensional kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M274">View MathML</a>. Hence, we are in position of applying the following result that has been more or less proved in [13].

Theorem 3.2Let, X, Ybe Banach spaces and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M288">View MathML</a>a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-map such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M290">View MathML</a>has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>, d-dimensional, manifold of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M292">View MathML</a>. Assume that for anyμin a neighborhood of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293">View MathML</a>the linearization<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M294">View MathML</a>has the null space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M295">View MathML</a>. Assume further that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M296">View MathML</a>is Fredholm with index zero and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M297">View MathML</a>a projection ofYonto the range of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M296">View MathML</a>. Then if the Poincaré-Melnikov function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M299">View MathML</a>

has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M301">View MathML</a>and a unique map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M302">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M303">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304">View MathML</a>is an isomorphism for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284">View MathML</a>.

Actually the statement in [[13], Theorem 4.1] is slightly different from the above. Hence, we give a proof of Theorem 3.2 in the Appendix.

We apply Theorem 3.2 to the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M204">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M307">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M308">View MathML</a> is independent of τ, and hence so is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M309">View MathML</a>. Next <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M310">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M311">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M314">View MathML</a>, is any vector satisfying (3.1). To apply Theorem 3.2, we look at the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M315">View MathML</a> with respect to ε at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a>. First, we have:

whereas differentiating (2.10) with respect to ε at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M319">View MathML</a>

We obtain then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M320">View MathML</a>

and then the Poincaré-Melnikov function is:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M321">View MathML</a>

(3.9)

The conclusion of Theorem 3.1 now easily follows from (3.9) and Theorem 3.2.  □

4 Poincaré-Melnikov function and adjoint system

In this section, we want to give a suitable definition of the adjoint system of the linearization of (1.6) along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M43">View MathML</a> in such a way that the Poincaré-Melnikov function (3.2) can be put in relation with the solutions of such an adjoint system.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M323">View MathML</a> be the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324">View MathML</a>-map defined in Introduction and recall the impact equation (1.6):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M325">View MathML</a>

(4.1)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, (4.1) has the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M328">View MathML</a>. We let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M329">View MathML</a> denote the solution of the impact system (4.1) on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M330">View MathML</a>. Then its derivative with respect to α at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a> satisfies the linearized equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M332">View MathML</a>

(4.2)

Next, recalling (1.1), we consider a perturbed impact system of (4.1) (see also (2.8)) of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M333">View MathML</a>

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M334">View MathML</a> is defined as follows: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M335">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M336">View MathML</a> is the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M337">View MathML</a>

Note that R is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-map on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M339">View MathML</a> taking values on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M341">View MathML</a>; moreover, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M342">View MathML</a> is autonomous then R is independent of τ, so we may take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343">View MathML</a> in its definition. We recall that for simplicity we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344">View MathML</a> instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M345">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M346">View MathML</a>.

To study the problem of existence of solutions of system (4.3), we are then led to find conditions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M347">View MathML</a>, d and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348">View MathML</a> so that the non-homogeneous linear equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M349">View MathML</a>

(4.4)

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M350">View MathML</a>. Let us comment on equation (4.4) (and similarly on (4.2)) that condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M351">View MathML</a> only involves the derivative of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344">View MathML</a> on the tangent space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M353">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M355">View MathML</a>. So, it is independent of any extension we take of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344">View MathML</a> to a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M357">View MathML</a>. We also note that for simplicity we denote again by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348">View MathML</a> the value of the linear functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M348">View MathML</a> in (4.4).

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M360">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M361">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M362">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M363">View MathML</a>

So, if equation (4.4) has a solution, we must necessarily have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M364">View MathML</a>

Next, we define two Hilbert spaces:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M365">View MathML</a>

Note Y is a Hilbert space and X is a closed subspace of a Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M366">View MathML</a>. Then (4.4) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M367">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M368">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M369">View MathML</a>.

Lemma 4.1The range<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M370">View MathML</a>is closed.

Proof Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M371">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M372">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M373">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M374">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M375">View MathML</a> is closed, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M376">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M377">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M378">View MathML</a>

By taking

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M379">View MathML</a>

we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M380">View MathML</a>. The proof is finished. □

Next, we prove the following result.

Proposition 4.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M381">View MathML</a>. Then the inhomogeneous system (4.4) has a solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M382">View MathML</a>if and only if equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M383">View MathML</a>

(4.5)

holds for any solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384">View MathML</a>of the adjoint system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M385">View MathML</a>

(4.6)

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M386">View MathML</a>.

Proof Before starting with the proof we observe that, because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M387">View MathML</a>, ψ is not uniquely determined by equation (4.5) since changing it with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M388">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M389">View MathML</a>, the equation remains the same. So, in equation (4.5), we look for ψ in a subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M390">View MathML</a> which is transverse to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M391">View MathML</a>. It turns out that the best choice, from a computational point of view, is to take ψ so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M392">View MathML</a> (see equation (3.1)).

First, we prove necessity. Assume that (4.4) can be solved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M393">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M394">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M395">View MathML</a>, be a solution of equation (4.6). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M396">View MathML</a>

Plugging these equalities in the left-hand side of (4.5) and integrating by parts, (4.5) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M397">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M398">View MathML</a>

(4.7)

because of the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282">View MathML</a> and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M400">View MathML</a> satisfies (4.6).

To prove the sufficiency, we show that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M381">View MathML</a> does not belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M370">View MathML</a>, then there exists a solution of the variational equation (4.6) such that (4.7) does not hold. So, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M403">View MathML</a>. By Lemma 4.1 and the Hahn-Banach theorem, there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M404">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M405">View MathML</a>

(4.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M406">View MathML</a>

(4.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M407">View MathML</a> is the usual scalar product on Y. We already noted that we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M408">View MathML</a>, and (4.8)-(4.9) remain valid. Repeating our previous arguments, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M409">View MathML</a> and that (4.8) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M410">View MathML</a> solves the adjoint system (4.6). Summarizing, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M403">View MathML</a> there exists a solution of the adjoint system for which (4.6) does not hold. This finishes the proof. □

Again we note that equation (4.6) only depends on the derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M412">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M413">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M414">View MathML</a>, where we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M392">View MathML</a> or, in other words, it is independent of any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324">View MathML</a>-extension we take of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M344">View MathML</a> to the whole <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M31">View MathML</a>.

We now prove the following proposition.

Proposition 4.2The adjoint system (4.6) has a solution if and only if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M419">View MathML</a>satisfy the first and the third equation in (3.1) (and we take the second equation in (3.1) as definition of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M282">View MathML</a>).

Proof Indeed let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384">View MathML</a> be a solution of (4.6) then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M422">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M423">View MathML</a> being the fundamental matrix of the linear equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M424">View MathML</a>. Then, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M425">View MathML</a> the two remaining condition in (4.6) read:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M426">View MathML</a>

that can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M427">View MathML</a>

or else, on account of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M428">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M429">View MathML</a>

The proof is finished. □

We conclude this section giving another expression of the Poincaré-Melnikov function (3.2) in terms of the solution of the adjoint system (4.6). To this end, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M384">View MathML</a> be a solution of the adjoint system (4.6). Since a fundamental matrix of the linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M431">View MathML</a>

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M432">View MathML</a> we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M433">View MathML</a>

so:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M434">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M435">View MathML</a>

As for the first term in the above equality, we can show it is related to the impact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M436">View MathML</a>. Indeed, from Section 2 we know that the solution of the singular equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M437">View MathML</a>

can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M438">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M439">View MathML</a> as in equation (2.6). Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M440">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M441">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M442">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M443">View MathML</a>

and then, using again <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M444">View MathML</a> we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M445">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M446">View MathML</a>

(4.10)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M342">View MathML</a> is autonomous, then R is independent of τ, and the expression (4.10) of the Poincaré-Melnikov function should be compared with the one given in [[14], Theorem 4.2] where a Poincaré-Melnikov function, characterizing transition to chaos, is given for almost periodic perturbations of autonomous impact equations with a homoclinic orbit.

5 The case of a manifold of periodic solutions

In this section we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M448">View MathML</a> for any α in (an open neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a> in) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20">View MathML</a>. Hence, from (3.3), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M451">View MathML</a>

We distinguish the two cases: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453">View MathML</a> for all α in (an open neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a> in) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20">View MathML</a>. First, we assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M456">View MathML</a>

Then a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M458">View MathML</a>-dimensional submanifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M459">View MathML</a> of (an open neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a> in) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M20">View MathML</a> exists such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463">View MathML</a>. So, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M272">View MathML</a> has the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M466">View MathML</a>-dimensional manifold of solutions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M467">View MathML</a>

So, we are in position to apply Theorem 3.2. First, we have to verify that the kernel <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M468">View MathML</a> equals the tangent space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M469">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M470">View MathML</a>, and then that the Poincaré-Melnikov function (vector):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M471">View MathML</a>

has a simple zero at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M472">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M473">View MathML</a>

From (3.3), we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M474">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475">View MathML</a> does not depend on τ. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M476">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M477">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463">View MathML</a> we easily see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M479">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M480">View MathML</a>. On the other hand, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M481">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M482">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M258">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M263">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M485">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M486">View MathML</a>

that, on account of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M477">View MathML</a> is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M488">View MathML</a>

Now, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M250">View MathML</a> we get, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M490">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M491">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M492">View MathML</a>

which, in turn, is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M493">View MathML</a> because of transversality and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M494">View MathML</a>.

Hence, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M495">View MathML</a>.

Now we consider the second condition. The Poincaré-Melnikov function (vector) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M496">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M463">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M498">View MathML</a>

(5.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M499">View MathML</a> is a matrix whose rows are left eigenvectors of zero eigenvalue of the matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M501">View MathML</a>

(5.2)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M502">View MathML</a> does not depend on τ since so does <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M475">View MathML</a>. Then (5.1) reads:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M504">View MathML</a>

Arguing as in Section 3, equation (5.2) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M505','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M505">View MathML</a>

(5.3)

Moreover, the adjoint variational system along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M506">View MathML</a> is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M507','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M507">View MathML</a>

(5.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M508">View MathML</a> satisfy equation (5.2). Then the Poincaré-Melnikov vector can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M509">View MathML</a>

(5.5)

or else

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M510">View MathML</a>

(5.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M511">View MathML</a> being the solution of (5.4) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M121">View MathML</a> the fundamental matrix of the linear equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M513">View MathML</a>

Of course the only difference between the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M516">View MathML</a> is that in the first case the Poincaré-Melnikov function is defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M517">View MathML</a> while in the second it is defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M518">View MathML</a> for an open neighborhood <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M519">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M520">View MathML</a>. Summarizing, we proved the following result.

Theorem 5.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M521">View MathML</a>for anyαin a neighborhood of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, and that either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453">View MathML</a>for anyα (in the same neighborhood). Then system (5.3) has ad-dimensional space of solutions where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M525">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M526">View MathML</a>according to which of the two conditions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M90">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M453">View MathML</a>holds. Moreover, if the Poincaré-Melnikov function (5.5) (or (5.6)) has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M529">View MathML</a>then system (1.1) has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M17">View MathML</a>-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M104">View MathML</a>satisfying (2.1).

Finally, we note that when we can show that a Brouwer degree of a Poincaré-Melnikov function from either Theorem 3.1 or 5.1 is non-zero then by following [15] we can show existence results.

6 Examples

We consider a second-order equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M532">View MathML</a>

with the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M533">View MathML</a> as discontinuity manifold (i.e., with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M534">View MathML</a>). We write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M535','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M535">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M536">View MathML</a> (i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M537">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M538','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M538">View MathML</a>). We also write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M539','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M539">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M540','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M540">View MathML</a>

i.e., we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M541','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M541">View MathML</a>

in the plane coordinates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M542">View MathML</a>. According to equation (5.4), the adjoint variational system reads, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M543">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M544','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M544">View MathML</a>

which can be written as (with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M545">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M546">View MathML</a>):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M547">View MathML</a>

(6.1)

Note that (when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M548">View MathML</a>) the last three equation are actually the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M543">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M550">View MathML</a> in terms of the unique (up to a multiplicative constant) bounded solution of the boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M551">View MathML</a>

and the Poincaré-Melnikov function (5.6) reads:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M552">View MathML</a>

whereas (4.10) reads:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M553">View MathML</a>

As an example, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M554">View MathML</a> that is we consider the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M555">View MathML</a>

The unperturbed equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M556">View MathML</a> with the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M557">View MathML</a> has the solutions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M558">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M559">View MathML</a>. Note that, to have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M560">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M561">View MathML</a> we need <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M562">View MathML</a>.

We assume we are in the first (non degenerate) case that is it holds (1.4), which now has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M563">View MathML</a>

(6.2)

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M564">View MathML</a> for this case. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M565">View MathML</a>

(6.2) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M566">View MathML</a>

(6.3)

Then it is easily seen that system (6.1), with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M568">View MathML</a>

Solving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M569">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M570">View MathML</a> and the boundary condition reads: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M571">View MathML</a>. So, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M572">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M573">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M574">View MathML</a>

and the Poincaré-Melnikov function reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M575">View MathML</a>

For example, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M576">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a> is the time the solution of equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M578">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M579">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M580">View MathML</a> takes to reach the discontinuity manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M533">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M582">View MathML</a>

which has a simple zero at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343">View MathML</a>.

To conclude the example we need to find a second-order equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M578">View MathML</a> such that (6.3) holds. We consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M585">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M586">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M587','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M587">View MathML</a>. It has the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M588">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M589">View MathML</a>. So, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M590">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M591">View MathML</a>. Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M592">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M593">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M594">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M595">View MathML</a>, so we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M596">View MathML</a>. Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M597">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M598">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M599">View MathML</a>

(6.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M600">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M601">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M602">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M603">View MathML</a>. Clearly, (6.4) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M604">View MathML</a>. Then the second solution is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M605">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M606">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M607">View MathML</a>

Consequently, if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M608">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M609">View MathML</a>. So, we conclude with the following.

Corollary 6.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M610">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M611">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M612">View MathML</a>be<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>functions such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M614">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M615">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M616">View MathML</a>

Suppose, also, that the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M617">View MathML</a>

has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a>, sufficiently small the singularly perturbed system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M620">View MathML</a>

has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M621">View MathML</a>-periodic solution orbitally near the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M622">View MathML</a>.

To get a second example, we change the above as follows: we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M623">View MathML</a>

with equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M624">View MathML</a>

It should be noted that the discontinuity line is the union of the two half lines <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M625">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M626">View MathML</a> which is not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324">View MathML</a>. However, all results hold true as long as we remain outside a (small) neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M628">View MathML</a>.

The unperturbed equation on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M176">View MathML</a> has the solutions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M630">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M631','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M631">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M632">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M633','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M633">View MathML</a> is the value of the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M634">View MathML</a> of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M635">View MathML</a>

at the time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M636">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M637">View MathML</a>. Since the equation has the Hamiltonian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M638">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M639">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M640">View MathML</a>

(6.5)

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M641','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M641">View MathML</a>

(6.6)

We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a> is the first positive time such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M643">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M644','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M644">View MathML</a> is the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M645">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M646">View MathML</a>

(6.7)

More related results are derived in the Appendix.

Then equations (6.2) have to be changed to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M647">View MathML</a>

(6.8)

But

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M648">View MathML</a>

and (6.8) easily follows. Now we compute the variational equation and the Poincaré-Melnikov function. From (6.6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M649">View MathML</a>, it follows that we can take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M650">View MathML</a>

from which we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M651">View MathML</a>

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M652">View MathML</a> has a homoclinic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M653">View MathML</a>, so the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M654">View MathML</a> is a part of a periodic solution inside of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M655','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M655">View MathML</a> bounded by the homoclinic one (see Figure 1). Then, since in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M656">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M657','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M657">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M658">View MathML</a>

Finally, since the equations on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M659">View MathML</a> can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M660">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M661">View MathML</a>

Putting all together we see that the adjoint variational system reads:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M662">View MathML</a>

The first three equations give the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M663">View MathML</a>

possessing the unique solution (up to a multiplicative constant) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M664">View MathML</a> which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M665">View MathML</a>

and, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M666">View MathML</a>, the Poincaré-Melnikov function is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M667">View MathML</a>

(6.9)

We conclude with the following.

thumbnailFigure 1. The upper parts of homoclinic and periodic orbits of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M668">View MathML</a>.

Corollary 6.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M128">View MathML</a>be as in equation (6.7), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670">View MathML</a>be a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M671">View MathML</a>-periodic, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>function and suppose that the function (6.9) has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343">View MathML</a>. Then, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M8">View MathML</a>, sufficiently small the singularly perturbed system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M675">View MathML</a>

has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M671">View MathML</a>-periodic solution orbitally near the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M677">View MathML</a>.

As an example of the second situation, we consider the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M678">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M679">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M680">View MathML</a>, i.e., we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M681">View MathML</a>

(6.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670">View MathML</a> is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M683">View MathML</a>-periodic, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a> function. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M685">View MathML</a>

we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M686">View MathML</a> for any α in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M687">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M688">View MathML</a>. Hence, we are in the degenerate case considered in Section 5. The adjoint variational equation along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M506">View MathML</a> reads now

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M690">View MathML</a>

The first two equations have the two-dimensional family of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M691">View MathML</a>. We take the two independent solutions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M692">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M693">View MathML</a> with the corresponding vectors:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M694">View MathML</a>

With <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M695','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M695">View MathML</a> (which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M696">View MathML</a> is independent of ε) the Poincaré-Melnikov vector is then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M697">View MathML</a>

Then we obtain the following corollary.

Corollary 6.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M670">View MathML</a>be a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M683">View MathML</a>-periodic, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>function and suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M701">View MathML</a>has a simple zero at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M702">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M343">View MathML</a>. Then the singularly perturbed system (6.10) has a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M621">View MathML</a>-periodic solution orbitally near the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M705">View MathML</a>.

Appendix

A.1 Further properties of the solution of (6.5)

From the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M706">View MathML</a>

we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M707">View MathML</a>

(7.1)

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M708">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M709">View MathML</a>

and, for α sufficiently small (in fact for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M710">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M711">View MathML</a>

Using formula 3.131.5 in [[16], p.254] we know that, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M712">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M713">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M714">View MathML</a>

and F is the elliptic integral of the first kind.

Next note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M715">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M716">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M717">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M718">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M637">View MathML</a>. Hence, (7.1) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M720">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M721','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M721">View MathML</a>

So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M722">View MathML</a>

We are interested in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M724">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M725">View MathML</a>

(7.2)

On the other hand, by (6.7), we directly verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M726','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M726">View MathML</a> by a numerical integration. But we derived (7.2) to get an explicit formula for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M727','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M727">View MathML</a> and in general for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M636">View MathML</a>.

Furthermore, the above computations also give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M729','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M729">View MathML</a>

(7.3)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M730','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M730">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M731','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M731">View MathML</a>

(7.4)

Solving (7.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M732','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M732">View MathML</a>

where am is the Jacobi amplitude function. Solving (7.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M733','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M733">View MathML</a>

(7.5)

for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M734','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M734">View MathML</a>

(7.6)

where cn is the Jacobi elliptic function. Formulas (7.5) and (7.6) give explicit solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M639">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M49">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M737','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M737">View MathML</a>

(7.7)

We can also compute the Taylor series of (7.7) integrating by series the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M652">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M739','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M739">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M740','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M740">View MathML</a>. Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M741','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M741">View MathML</a>

we see that the following recurrence condition holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M742','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M742">View MathML</a>

where we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M743','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M743">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M744','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M744">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M745">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M745','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M745">View MathML</a>, we see by the induction that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M747','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M747">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M748','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M748">View MathML</a> (note that in the product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M749','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M749">View MathML</a> one of the two indexes is odd). So,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M750','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M750">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M751','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M751">View MathML</a>

For the first few indexes, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M752','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M752">View MathML</a>

so that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M753','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M753">View MathML</a>

On the other hand, using Mathematica, we can expand (7.7) to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M754','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M754">View MathML</a>

which coincides with our above analytical expansion.

A.2 Proof of Theorem 3.2

Here, we prove Theorem 3.2. We emphasize the fact that proof mainly follows the idea in [[13], Theorem 4.1].

Proof of Theorem 3.2 The existence part is quite standard so we sketch it and give emphasis to the proof of invertibility of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M757','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M757">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M758','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M758">View MathML</a> and, differentiating twice, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M759','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M759">View MathML</a>. As a consequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M760','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M760">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M761','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M761">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M762','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M762">View MathML</a> as in the statement of the theorem. We write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M763','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M763">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M764','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M764">View MathML</a>. Applying the Implicit Function Theorem to the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M765','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M765">View MathML</a>, we get the existence of a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M3">View MathML</a>-solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M767','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M767">View MathML</a> of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M768','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M768">View MathML</a>. From uniqueness, we obtain also

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M769','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M769">View MathML</a>

Next, differentiating the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M770','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M770">View MathML</a> with respect to μ and to ε at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M771','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M771">View MathML</a>, we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M772','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M772">View MathML</a>

Next, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284">View MathML</a>, equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M774','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M774">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M775','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M775">View MathML</a>, but the l.h.s. tends, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M110">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M777','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M777">View MathML</a> which gives the Poincaré-Melnikov condition. We conclude that, if the Poincaré-Melnikov condition is satisfied, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M778','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M778">View MathML</a> (small) there exists a unique solution of equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M779','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M779">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M780','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M780">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M781','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M781">View MathML</a>.

Now we prove the invertibility of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M782','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M782">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M304">View MathML</a> is Fredholm with index zero, it is enough to prove that equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M784','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M784">View MathML</a> has, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M284">View MathML</a>, the unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M786','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M786">View MathML</a>. Although <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M787','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M787">View MathML</a> is only <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M324">View MathML</a> with respect to ε, it is linear in z. Thus, we can still apply the existence and uniqueness argument given above. Of course, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M789','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M789">View MathML</a> vanishes on the linear subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M790','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M790">View MathML</a>, and clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M791','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M791">View MathML</a>. Next <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M792','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M792">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M793','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M793">View MathML</a>. Thus, from the existence and uniqueness result it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M794','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M794">View MathML</a> if the following condition is satisfied:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M795','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M795">View MathML</a>

On account of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M796','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M796">View MathML</a> we are led to look at the solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M797','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M797">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M798">View MathML</a>. From the previous remarks, we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M799','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M799">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M798','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M798">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M801','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M801">View MathML</a>. So, the claim to be proved is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M802','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M802">View MathML</a>

where we have replaced z with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M803','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M803">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M804','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M804">View MathML</a>. Now we differentiate the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M805','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M805">View MathML</a>

with respect to μ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M293">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M807','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M807">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M808','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M808">View MathML</a>

since, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M809','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M809">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M810','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M810">View MathML</a> we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M811','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/71/mathml/M811">View MathML</a>. The proof of Theorem 3.2 is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The work presented here was carried out in collaboration between the authors. The authors contributed to every part of this study equally and read and approved the final version of the manuscript.

Acknowledgements

This article is dedicated to Professor Jean Mawhin on the occasion of his 70th birthday.

BF is partially supported by GNAMPA-CNR and MURST-group ‘Equazioni differenziali ordinarie e applicazioni’ (Italy). MF is partially supported by the grant APVV-0134-10 and Marche Polytechnic University, Ancona (Italy).

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