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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Melnikov theory for weakly coupled nonlinear RLC circuits

Flaviano Battelli1* and Michal Fečkan23

Author Affiliations

1 Department of Industrial Engineering and Mathematical Sciences, Marche Polytecnic University, Via Brecce Bianche 1, Ancona, 60131, Italy

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

3 Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, Bratislava, 814 73, Slovakia

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Boundary Value Problems 2014, 2014:101  doi:10.1186/1687-2770-2014-101


Dedicated to Professor Ivan Kiguradze


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/101


Received:18 December 2013
Accepted:1 April 2014
Published:7 May 2014

© 2014 Battelli and Fe¿kan; 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 credited.

Abstract

We apply dynamical system methods and Melnikov theory to study small amplitude perturbation of some coupled implicit differential equations. In particular we show the persistence of such orbits connecting singularities in finite time provided a Melnikov like condition holds. Application is given to coupled nonlinear RLC system.

MSC: 34A09, 34C23, 37G99.

Keywords:
implicit ode; perturbation; Melnikov method; RLC circuits

1 Introduction

In [1], motivated by [2,3], the equation modeling nonlinear RLC circuits

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M1">View MathML</a>

(1)

has been studied. It is assumed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M3">View MathML</a> are smooth functions with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M2">View MathML</a> at least quadratic at the origin and satisfying suitable assumptions. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M5">View MathML</a> the equation reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M6">View MathML</a>

(2)

It is assumed that, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M7">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M9">View MathML</a>. So for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a> (2) has the Hamiltonian

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M11">View MathML</a>

passing through <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12">View MathML</a>. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M13">View MathML</a> and the Hessian of ℋ at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M14">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M15">View MathML</a>

so that the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M9">View MathML</a> means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12">View MathML</a> is a saddle for ℋ. Multiplying the second equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M18">View MathML</a> we get the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M19">View MathML</a>

(3)

Note that (2) falls in the class of implicit differential equations (IODE) like

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M20">View MathML</a>

(4)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M21">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M22">View MathML</a> vanishes on the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M23">View MathML</a> and the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M24">View MathML</a> implies that the line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M25">View MathML</a> consists of noncritical 0-singularities for (3) (see [[4], p.163]). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M26">View MathML</a> denote the kernel of the linear map L and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M27">View MathML</a> its range. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M28">View MathML</a> is the subspace having zero first component and then the right hand side of (3) belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M28">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M30">View MathML</a>. So all the singularities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M23">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M32">View MathML</a> are impasse points while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M12">View MathML</a> is a so called I-point (see [[4], pp.163-166]). Quasilinear implicit differential equations, such as (4), find applications in a large number of physical sciences and have been studied by several authors [4-12]. On the other hand, there are many other works on implicit differential equations [13-18] dealing with more general implicit differential systems by using analytical and topological methods.

Passing from (2) to (3), in the general case, it corresponds to multiplying (4) by the adjugate matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M34">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M36">View MathML</a>. Here we note that A and x may have different dimensions in this paper depending on the nature of the equation but the concrete dimension is clear from that equation, so we do not use different notations for A and x. Basic assumptions in [1] are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M40">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a> (that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a> is an I-point for (4)) and the existence of a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43">View MathML</a> in a bounded interval J tending to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a> as t tends to the endpoints of J.

It is well known [4,8] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38">View MathML</a> imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M47">View MathML</a>

(5)

and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M39">View MathML</a> is equivalent to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M49">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M50">View MathML</a>. It has been proved in [19] that (5) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M51">View MathML</a> is at most 2. So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M52">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M53">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M54">View MathML</a> cannot be hyperbolic for the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M55">View MathML</a>.

In this paper we study coupled IODEs such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M56">View MathML</a>

(6)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M59">View MathML</a> and other assumptions that will be specified below. Let us remark that (6) is a special kind of the general equation (4) with, among other things,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M60">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M61">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M64">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65">View MathML</a> is not a I-point. Multiplying the first equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M66">View MathML</a> and the second by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M67">View MathML</a> we obtain the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M68">View MathML</a>

(7)

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M71">View MathML</a> satisfy the following assumptions:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M73">View MathML</a> and the unperturbed equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M74">View MathML</a>

(8)

possesses a noncritical singularity at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a>, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M38">View MathML</a>.

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M78">View MathML</a> and the spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M79">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M80">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M81">View MathML</a>

has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82">View MathML</a> homoclinic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M84">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M85">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a>. Without loss of generality, we may, and will, assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M87">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M90">View MathML</a> are 1-periodic in t with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M91">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M92">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M93">View MathML</a> and ε sufficiently small.

(C3) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M94">View MathML</a> be the eigenvectors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M95">View MathML</a> with the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M96">View MathML</a>, resp. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M97">View MathML</a> (or else <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M98">View MathML</a>).

From (C2) we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M99">View MathML</a> is a bounded solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M100">View MathML</a>

(9)

and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a> persists as a singularity of (7). So this paper is a continuation of [1,19], but here we study more degenerate IODE.

The objective of this paper is to give conditions, besides (C1)-(C3), assuring that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M102">View MathML</a>, the coupled equations (7) has a solution in a neighborhood of the orbit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M103">View MathML</a> and reaching <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65">View MathML</a> is a finite time. Our approach mimics that in [1] and uses Melnikov methods to derive the needed conditions. Let us briefly describe the content of this paper. In Section 2 we make few remarks concerning assumptions (C1)-(C3). Then, in Section 3, we change time to reduce equation (7) to a smooth perturbation of (9) whose unperturbed part has the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M105">View MathML</a>. Next, in Section 4 we derive the Melnikov condition. Finally Section 5 is devoted to the application of our result to coupled equations of the form (1) for RLC circuits, while some computations are postponed to the appendix.

We emphasize the fact that Melnikov technique is useful to predict the existence of transverse homoclinic orbits in mechanical systems [20,21] together with the associated chaotic behavior of solutions. However, the result in this paper is somewhat different in that we apply the method to show existence of orbits connecting a singularity in finite time.

2 Comments on the assumptions

By following [1,19] we note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M106">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M107">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M108">View MathML</a> is a bounded solution of the linear equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M109">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M110">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M111">View MathML</a>. We get then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M113">View MathML</a>

So

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M114">View MathML</a>

From [[22], Theorem 4.3, p.335 and Theorem 4.5, p.338] it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M115">View MathML</a>

(10)

and there exist a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M116">View MathML</a> and a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M117">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M118">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M119">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M120">View MathML</a> since otherwise <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M121">View MathML</a>, contradicting (10). Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M122">View MathML</a> is an eigenvector of the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M123">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M95">View MathML</a>. We have then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M125">View MathML</a>

for a suitable constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M126">View MathML</a>. As a consequence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M127">View MathML</a>

Next

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M128">View MathML</a>

Taking logarithms, dividing by s and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M130">View MathML</a>

that is, in (10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M131">View MathML</a> can be replaced with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M132">View MathML</a>. Similarly, changing s with −s:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M133">View MathML</a>

Next, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M134">View MathML</a>

(11)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M135">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M137">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M138">View MathML</a> we have then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M139">View MathML</a>

(12)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M140">View MathML</a>

From (C2), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M141">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M143">View MathML</a>. Hence condition (C3) means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82">View MathML</a> tends transversally to the singular manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M145">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a>.

As in [19] it is easily seen that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M147">View MathML</a>

(13)

and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148">View MathML</a> solves the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M149">View MathML</a>

(14)

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148">View MathML</a> is a bounded solution of (14). Next, setting as in [19]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M151">View MathML</a>

(15)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M152">View MathML</a>, it is easily seen that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M153">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M154">View MathML</a> whose linearization along <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M153">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M156">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M157">View MathML</a>

(16)

Note, then, that (14) is derived from (16) with the change <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M158">View MathML</a>. This fact should clarify why we need to consider the linear system (14) instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M159">View MathML</a>. However, see [19] for a remark concerning the space of bounded solutions of (14) and that of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M159">View MathML</a>.

We now prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148">View MathML</a> is the unique solution of equation (14) which is bounded on ℝ. This is a kind of nondegeneracy of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82">View MathML</a>.

Lemma 2.1Assume (C2) and (C3) hold. Then, up to a multiplicative constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M148">View MathML</a>is the unique solution of (14) which is bounded on ℝ.

Proof From [[19], Lemma 3.1] it follows that the linear map:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M164">View MathML</a>

has the simple eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M166">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M167">View MathML</a>, then the linear map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M168">View MathML</a>

has the eigenvalues ±μ; moreover, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M169">View MathML</a>

for two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M170">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M171">View MathML</a> is a solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M172">View MathML</a>

(17)

satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M173">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M174">View MathML</a>. Then (17) satisfies the assumptions of [[19], Theorem 5.3] and hence its conclusion with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M175">View MathML</a>, that is, the fundamental matrix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M176">View MathML</a> of (17) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M177">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M178">View MathML</a>. However, it is well known (see [23-25]) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M179">View MathML</a> is the space of initial conditions for the bounded solutions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M180">View MathML</a> of (17) that, then, tend to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129">View MathML</a> at the exponential rate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M182">View MathML</a>. As a consequence a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M183">View MathML</a> of (17) is bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M184">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M185">View MathML</a> is a bounded solution of (14). Then we conclude that the space of solutions of (14) that are bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M180">View MathML</a> is one dimensional.

Incidentally, since the fundamental matrix of (14) is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M187">View MathML</a>, we note that it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M188">View MathML</a>

Using a similar argument in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M189">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M190">View MathML</a>, and [[19], Theorem 5.4] instead of [[19], Theorem 5.3] with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M191">View MathML</a> we see that (14) has at most a one dimensional space of solutions bounded in ℝ. More precisely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M192">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M193">View MathML</a>, and a projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M194">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M195">View MathML</a> exists such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M196">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M197">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M198">View MathML</a> is a solution of (14) bounded on ℝ we deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M199">View MathML</a> and the result follows. □

We conclude this section with a remark about condition (c) in [[19], Theorem 5.3]. Consider a system in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M200">View MathML</a> such as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M201">View MathML</a>

(18)

Then the following result holds.

Theorem 2.2Suppose the following hold:

(i) Dhas two simple eigenvalues<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M202">View MathML</a>and all the other eigenvalues ofDhave either real part less than<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a>or greater than<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M205">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M206">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M129">View MathML</a>.

Then there are as many solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43">View MathML</a>of (18) satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M209">View MathML</a>

(19)

as the dimension of the space of the generalized eigenvectors of the matrixDwith real parts less than or equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a>; here<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M211">View MathML</a>are two suitable positive constants. Similarly there are as many solutions of (18) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M212">View MathML</a>

(20)

for suitable constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M213">View MathML</a>, as the dimension of the space of the generalized eigenvectors of the matrixDwith real parts greater than or equal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a>.

Proof We prove the first statement concerning (19). By a similar argument (20) is handled. Changing variables we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M215">View MathML</a>

and the eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M216">View MathML</a> have real parts less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a> and those of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M218">View MathML</a> have real parts greater than or equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a>. So the system reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M220">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M221">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M222">View MathML</a> are matrices (or vectors) of suitable orders. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M223">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M224">View MathML</a>

(22)

Now we observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225">View MathML</a> is a solution of (22) bounded at +∞ if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M43">View MathML</a> is a solution of (21) which is bounded on ℝ when multiplied by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M227">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M229">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M230">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M231">View MathML</a>, the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M232">View MathML</a> exists for any solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225">View MathML</a> of (22) bounded on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234">View MathML</a>. So, let us fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M235">View MathML</a> and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M236">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225">View MathML</a> is a solution of (22) bounded at +∞ it must be, by the variation of constants formula,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M238">View MathML</a>

(23)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M240">View MathML</a>. Note that since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M242">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M222">View MathML</a> are bounded, we can interpret (22) as a fixed point theorem in the Banach space of bounded function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M245">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M246">View MathML</a>

with the obvious norm. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M247">View MathML</a> we see that the map (23) is a contraction on B, provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M248">View MathML</a> is sufficiently large, and then, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M249">View MathML</a>, it has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M250">View MathML</a>. Note that a priori<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M251">View MathML</a> is defined only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M245">View MathML</a> but of course we can extend it to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M253">View MathML</a> going backward with time. We now prove that positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M254">View MathML</a> exist such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M255">View MathML</a> fox any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M256">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M257">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M258">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M259">View MathML</a>

So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M116">View MathML</a> let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M261">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M262">View MathML</a> and set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M263">View MathML</a>. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M264">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M265">View MathML</a>. Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M266">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M267">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M268">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M269">View MathML</a>, from the above it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M270">View MathML</a>. Similarly we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M271">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M272">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M273">View MathML</a>. As a consequence we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M274">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M275">View MathML</a>

So, provided we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M276">View MathML</a> we see that eventually (i.e. for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M277">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M278">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M279">View MathML</a>

and the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M280">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M281">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M256">View MathML</a> follows from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M283">View MathML</a> cannot vanish in any bounded interval. Finally since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M284">View MathML</a> we get, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M285">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M286">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M287">View MathML</a>

The proof is complete. □

Remark 2.3 (i) It follows from the proof of Theorem 2.2 that inequalities of (19) also hold replacing (i) with the weaker assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a> is a simple eigenvalue of D and all the others either have real parts less than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M290">View MathML</a> (i.e. we do not need that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a> is simple). Similarly inequalities of (20) hold if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a> is a simple eigenvalue of D and all the others either have real parts greater than <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M204">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M294">View MathML</a> (i.e. we do not need that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M203">View MathML</a> is simple).

(ii) Note that a result related to Theorem 2.2 has been proved in [26].

3 Solutions asymptotic to the fixed point

It follows from (11)-(12) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M296">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M297">View MathML</a> then, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M298">View MathML</a> we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M299">View MathML</a>. Furthermore, from (13) we also get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M300">View MathML</a>

As a consequence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M301">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M141">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M303">View MathML</a> is a strictly increasing diffeomorphism (see (15) for the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M304">View MathML</a>). Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M305">View MathML</a> satisfies (8) on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M306">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M307">View MathML</a>

Moreover (see (13))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M308">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M41">View MathML</a> is not an I-point of (8). In this paper we want to look for solutions of the coupled equation (7) that belong to a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M310">View MathML</a>, they are defined in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M311">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312">View MathML</a>, and tend to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65">View MathML</a> at the same rate as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M314">View MathML</a>. To this end we first perform a change of the time variable as follows. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M315">View MathML</a>

and plug <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M316">View MathML</a> in (7). We get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M317">View MathML</a>

(24)

Since we are looking for solutions of (7) tending to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M65">View MathML</a> at the same rate as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M82">View MathML</a>, in (24) we make the change of variables

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M320">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M321">View MathML</a> is the bounded function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M322">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M323">View MathML</a>

(26)

for a suitable constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M324">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M326">View MathML</a> we get, using (C3), (26):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M327">View MathML</a>

(27)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M328">View MathML</a> large and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M329">View MathML</a> sufficiently small. Then (27) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M330">View MathML</a> imply the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M331">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M332">View MathML</a> so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M333">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M335">View MathML</a>. Now plugging (25) into (24) we derive the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M336">View MathML</a>

(28)

From (11) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M337">View MathML</a>

Next we note that from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M338">View MathML</a> it follows that the quantities

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M339">View MathML</a>

are bounded uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M342">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M343">View MathML</a>, ε bounded.

The linearization of (28) at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a> is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M346">View MathML</a>

(29)

Taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M347">View MathML</a> we get the systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M348">View MathML</a>

(30)

Similarly taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M138">View MathML</a> we get the systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M350">View MathML</a>

(31)

From the proof of Lemma 2.1 (see also [[1], Lemma 3.1]) we know that (30) has the positive simple eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M351">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M166">View MathML</a>, and (31) has the negative simple eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M353">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M354">View MathML</a>. From the roughness of exponential dichotomies it follows that both equations in (29) have an exponential dichotomy on both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M356">View MathML</a> with projections, resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M358">View MathML</a>. Hence (see also [19]) all solutions of the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M359">View MathML</a>

(32)

adjoint to (29), are bounded as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M360">View MathML</a>. We let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M361">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M362">View MathML</a> be any two linearly independent solutions of (32).

4 Melnikov function and the original equation

In this section we will give a condition for solving (28) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M363">View MathML</a> near the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M364">View MathML</a> of the same equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a>. Writing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M366">View MathML</a>

(33)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M367">View MathML</a>

(34)

we look for solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M368">View MathML</a> of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M369">View MathML</a>

(35)

in the Banach space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-functions on ℝ, bounded together with their derivatives and with small norms. We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371">View MathML</a> and equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M372">View MathML</a> reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M373">View MathML</a>

(36)

In Section 3 (see also [1,19]) we have seen that (36) has an exponential dichotomy on both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234">View MathML</a> with projections <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M376">View MathML</a>. So the only bounded solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M225">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M372">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M379">View MathML</a>. In other words <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M380">View MathML</a>. So we are lead to prove the following.

Theorem 4.1LetY, Xbe Banach spaces, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M381">View MathML</a>be a small parameter and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M382">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M383">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M384">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M385">View MathML</a>be<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386">View MathML</a>-functions such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M380">View MathML</a>;

(c) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M389">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M390">View MathML</a>.

Set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M391">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M392">View MathML</a>

(37)

and suppose there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M393">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M394">View MathML</a>and the derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M395">View MathML</a>is invertible. Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M396">View MathML</a>and unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398">View MathML</a>, defined in a neighborhood of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M399">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M400">View MathML</a>

(38)

and for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M402">View MathML</a>, (35) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M403">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M404">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M405">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M406">View MathML</a>-functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M407">View MathML</a>and we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M408">View MathML</a>

(39)

Proof We look for solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M409">View MathML</a> of (35) that are close to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M410">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M411">View MathML</a> be the projection such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M412">View MathML</a>. Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M413">View MathML</a>. From the implicit function theorem, we solve the projected equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M414">View MathML</a>

for unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M415">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M416">View MathML</a>

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M417">View MathML</a> is sufficiently small and η in a fixed closed ball <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M418">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M419">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M420">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386">View MathML</a>-smooth. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M422">View MathML</a>, we need to solve the bifurcation equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M423">View MathML</a>

(40)

Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M424">View MathML</a>

(41)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M425">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M426">View MathML</a>

uniformly with respect to η. We conclude that (40) can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M427">View MathML</a>

(42)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M428">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M429">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M386">View MathML</a>-functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M431">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M432">View MathML</a> uniformly with respect to η, so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M433">View MathML</a>

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M431">View MathML</a>. By (41), system (42) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M436">View MathML</a>

(43)

Because of the assumptions we can apply the implicit function theorem to (43) to obtain a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M438">View MathML</a> defined in a neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a> satisfying (43) and such that (38) holds. Setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M440">View MathML</a>

we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M441">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M442">View MathML</a> are bounded <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-solutions of (35) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M398">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M445','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M445">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M446">View MathML</a>. Then we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M447">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M448','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M448">View MathML</a>) for continuous <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M449">View MathML</a> where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M450">View MathML</a>

Clearly (39) follows differentiating the above equality at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a>. The proof is complete. □

Remark 4.2 Note that, because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M394">View MathML</a>, (39) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M453">View MathML</a>

which has the unique solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M454','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M454">View MathML</a>

(44)

Now we apply Theorem 4.1 to (28) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M455">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M456">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M457">View MathML</a> as in (33), (34) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M458">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M459">View MathML</a> is the Banach space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M460">View MathML</a>-functions bounded together with their derivatives with the usual sup-norm.

We already observed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M371">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M462">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M463">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M464">View MathML</a> have been defined in the previous Section 3. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M465">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466">View MathML</a>. We recall, from [19], that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M467">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M468">View MathML</a> are solutions of the adjoint equation of (16):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M469">View MathML</a>

(45)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M470">View MathML</a>. Hence (37) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M471">View MathML</a>

or passing to time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M472">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M473">View MathML</a>

(46)

A direct application of Theorem 4.1 gives the following.

Theorem 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475">View MathML</a>be given as in (46) where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477">View MathML</a>are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M478">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M479">View MathML</a>exist so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M480">View MathML</a>

(47)

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M481">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M482">View MathML</a>, unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M484">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M485">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M486">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M487">View MathML</a>, defined for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M488">View MathML</a>, and a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M489">View MathML</a>of (24) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M491">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M492">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M493">View MathML</a>

(48)

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M494">View MathML</a>

Remark 4.4 (i) Equation (48) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M495">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M406">View MathML</a>-functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M497">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M498','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M498">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M499">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M500">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M501','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M501">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M502">View MathML</a>

(49)

which gives a first order approximation of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M503">View MathML</a>. Next, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M504">View MathML</a> can be computed using (44) adapted to this case. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M504','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M504">View MathML</a> are bounded solutions of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M506','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M506">View MathML</a>

Since (36) has exponential dichotomies on both <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M234">View MathML</a> (with projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M357">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M356">View MathML</a> (with projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M358">View MathML</a>) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M511">View MathML</a>

(50)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M512">View MathML</a> is the fundamental solution of (36). Note formulas (50) are well defined at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M513">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M514">View MathML</a>, due to the first assumption of (47). Next, passing to time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M515">View MathML</a> and taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M516">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M517">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M518','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M518">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M519">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M520">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M521">View MathML</a> solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M522">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M523">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M524">View MathML</a> is a fundamental solution of (16).

(ii) Using (49), the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M525">View MathML</a> are bounded solutions of (7) in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M526','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M526">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M527">View MathML</a>

Summarizing, we obtain the following corollary of Theorem 4.3.

Corollary 4.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M466">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475">View MathML</a>be given as in (46) where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477">View MathML</a>are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M478">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M479">View MathML</a>exist so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M534">View MathML</a>

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M481">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M482">View MathML</a>, unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M370">View MathML</a>-functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M484">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M485">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M486">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M487">View MathML</a>, defined for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M488">View MathML</a>, and a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M543">View MathML</a>of (7) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M545">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M492">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M547">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M548','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M548">View MathML</a>

5 Applications to RLC circuits

In this section we study the coupled equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M549','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M549">View MathML</a>

(51)

which is motivated by [2,3]. Note that (51) is obtained by coupling two equations modeling nonlinear RLC circuits as in (1). Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M550">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M551">View MathML</a>, λ, χ and ϖ are positive parameters. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M552">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M553">View MathML</a>, (51) reads

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M554','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M554">View MathML</a>

(52)

By solving the second and fourth equations of (52) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M555">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M556">View MathML</a>, we get:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M557">View MathML</a>

(53)

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M558">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M559','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M559">View MathML</a>, to write the system (52) in the form (7) with parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M560">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M561">View MathML</a> fixed, we have to multiply the second and fourth equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M562','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M562">View MathML</a>, respectively, and we obtain the system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M563">View MathML</a>

(54)

with (uncoupled) unperturbed equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M10">View MathML</a> (see (8)):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M565">View MathML</a>

(55)

Neglecting left multiplicators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M566">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M567">View MathML</a>) in (55), we obtain the following system (see condition (C2)):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M568">View MathML</a>

(56)

Clearly, condition (C1) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M569">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M570">View MathML</a>

The equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M571">View MathML</a> has the prime integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M572">View MathML</a>. A solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M573">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M574">View MathML</a> has to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M575">View MathML</a> with the solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M576">View MathML</a>

bounded on ℝ. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M577">View MathML</a>

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M578">View MathML</a>. From

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M579','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M579">View MathML</a>

we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M580">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M581','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M581">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M582','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M582">View MathML</a>, we derive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M583">View MathML</a>, and condition (C3) holds as well. Next, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M584">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M91">View MathML</a> and assumption (C2) is also verified. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M586">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M90">View MathML</a>. Furthermore by (15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M588">View MathML</a>

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M589">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M590">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M591">View MathML</a>

So now (16) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M592">View MathML</a>

(57)

which has the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M593">View MathML</a>. In other words, we deal with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M594">View MathML</a>

(58)

possessing the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M595','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M595">View MathML</a>. Following [[27], p.327], the second solution of (58) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M596">View MathML</a>

Consequently a fundamental matrix solution of (57) has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M597">View MathML</a>

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M598','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M598">View MathML</a>. The adjoint system of (57) is (see (45))

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M599','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M599">View MathML</a>

(59)

with the fundamental matrix solution

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M600','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M600">View MathML</a>

Note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M601">View MathML</a>. Thus we take

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M602">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M603','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M603">View MathML</a>

We now compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M475">View MathML</a>. We have (see the appendix)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M605">View MathML</a>

(60)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M606">View MathML</a> is the sine integral function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M607">View MathML</a> is the cosine integral function [[28], p.886] and Γ is the Euler constant. Similarly

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M608','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M608">View MathML</a>

(61)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M609">View MathML</a>

(62)

Note

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M610">View MathML</a>

Consequently, the Melnikov function is now

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M611','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M611">View MathML</a>

(63)

The equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M612">View MathML</a> is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M613">View MathML</a>

(64)

So having <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M614">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M615">View MathML</a> such that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M616">View MathML</a>

(65)

has a simple zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M619">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M620">View MathML</a>, formulas (64) give a simple zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M621">View MathML</a> of (63) with positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M623">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624">View MathML</a>, and Corollary 4.5 can be applied to (51). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M625">View MathML</a>, then (65) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M626','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M626">View MathML</a>

(66)

Hence assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M627">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M628">View MathML</a>, the right hand side of (66) is negative, and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M629">View MathML</a>

(67)

satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M619">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617">View MathML</a> satisfies (65) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M618">View MathML</a>, condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M634','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M634">View MathML</a> is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M635">View MathML</a>. Then using (66), we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M636','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M636">View MathML</a>

(68)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M637','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M637">View MathML</a>, then (68) is not satisfied, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M638','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M638">View MathML</a>. So we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M639','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M639">View MathML</a> and (68) gives also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M640">View MathML</a>. Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M640','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M640">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M642','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M642">View MathML</a>.

Summarizing we see that for any fixed χ and ϖ satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M643','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M643">View MathML</a>

(69)

the Melnikov function (63) has a simple zero <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M621">View MathML</a> given by (64) and (67), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M645">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M646">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M647">View MathML</a>. Hence in the region given by (69) we apply Corollary 4.5 to (51) with parameters <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M550">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M551','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M551">View MathML</a>, λ near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M651','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M651">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624">View MathML</a> determined by (64) and (67), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M653','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M653">View MathML</a>

(70)

for any fixed ϖ and χ satisfying (69). Summarizing, we get the following.

Theorem 5.1For any fixedϖ, χsatisfying (69) and then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M617">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M622">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M623">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M624">View MathML</a>given by (67) and (70), there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M658','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M658">View MathML</a>and smooth functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M659">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M660">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M661','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M661">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M662">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M663">View MathML</a>, such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M664">View MathML</a>, system (51) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M665">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M666">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M667','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M667">View MathML</a>, possesses a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M668','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M668">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M669">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M670">View MathML</a>

(71)

Proof We apply Corollary 4.5 to (53). Since, in this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M671','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M671">View MathML</a>, according to Corollary 4.5 (53) has a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M672','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M672">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M553">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M674">View MathML</a>

(72)

Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M675','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M675">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M676">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M677','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M677">View MathML</a>. So, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M678','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M678">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M679">View MathML</a>

and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M680','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M680">View MathML</a>

(73)

or, equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M681">View MathML</a>

(74)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M682">View MathML</a> we see that, for ε sufficiently small,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M683">View MathML</a>

and hence, using (74) and (72),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M684">View MathML</a>

(75)

Vice versa, if (75) and the first of (72) hold, then (73) gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M685">View MathML</a>

Hence (71) and (72) are equivalent. The proof is complete. □

Of course, solutions given by Theorem 5.1 vary smoothly with respect <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M686">View MathML</a> satisfying (69).

Remark 5.2 Missed in the above analysis is the second possibility when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M687">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M688','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M688">View MathML</a> and (66) is negative if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M689','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M689">View MathML</a>

(76)

so we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M690','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M690">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M691">View MathML</a>. Then inequality of (68) is satisfied since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M692">View MathML</a>. So we conclude that the result of Theorem 5.1 is valid also for

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M693','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M693">View MathML</a>

Appendix

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M694','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M694">View MathML</a> be the second component of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M476">View MathML</a>. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M696','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M696">View MathML</a> is an even function and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M697','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M697">View MathML</a>

(77)

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M698">View MathML</a>

(78)

Now, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M699','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M699">View MathML</a> and integration by parts of the second integral,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M700">View MathML</a>

Furthermore, since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M701">View MathML</a>

we compute

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M702">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M703">View MathML</a>

Next

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M704','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M704">View MathML</a>

Furthermore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M705','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M705">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M706','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M706">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M707','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M707">View MathML</a>

Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M708','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M708">View MathML</a>. Next, using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M709','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M709">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M710','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M710">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M711','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M711">View MathML</a>, we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M712','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M712">View MathML</a>

Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M713','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M713">View MathML</a> we see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M714','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M714">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M715','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M715">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M716','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M716">View MathML</a> and using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M717','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M717">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M718','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M718">View MathML</a>

Summarizing we arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M719','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M719">View MathML</a>

So inserting the above computations into (77) and (78), we get (60) and the first formula of (62). Next, since the second component <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M720','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M720">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M477">View MathML</a> is odd, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M722','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M722">View MathML</a>

But

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M723','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M723">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M724','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M724">View MathML</a>

So we obtain (61). Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M725','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/101/mathml/M725">View MathML</a>

which implies the second formula of (62).

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

BF is partially supported by PRIN-MURST ‘Equazioni Differenziali Ordinarie e Applicazioni’ (Italy). MF is partially supported by the Grants VEGA-MS 1/0071/14, VEGA-SAV 2/0029/13 and Marche Polytechnic University, Ancona (Italy).

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