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

Open Access Research

Global continuation of periodic solutions for retarded functional differential equations on manifolds

Pierluigi Benevieri12, Alessandro Calamai3, Massimo Furi1 and Maria Patrizia Pera1*

Author Affiliations

1 Dipartimento di Matematica e Informatica, Università degli Studi di Firenze, Via S. Marta 3, Firenze, I-50139, Italy

2 Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo, 05508-090, Brasil

3 Dipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, Ancona, I-60131, Italy

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Boundary Value Problems 2013, 2013:21  doi:10.1186/1687-2770-2013-21

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/21


Received:19 October 2012
Accepted:18 January 2013
Published:11 February 2013

© 2013 Benevieri et al.; licensee Springer

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

Abstract

We consider T-periodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations.

Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitz-type global bifurcation result and a continuation principle of Mawhin type.

MSC: 34K13, 34C40, 37C25, 70K42.

Keywords:
retarded functional differential equations; global bifurcation; fixed point index; degree of a vector field

1 Introduction

In this paper we prove a global continuation result for periodic solutions of the following retarded functional differential equation (RFDE for short) on a manifold, depending on a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M2">View MathML</a>

(1.1)

Let us present the setting of the problem. Consider a boundaryless smooth m-dimensional manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M3">View MathML</a> and, given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M5">View MathML</a> stand for the tangent space of M at p. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M6">View MathML</a> the set of bounded and uniformly continuous maps from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M7">View MathML</a> into M, and observe that this is a metric space as a subset of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M8">View MathML</a> with the usual supremum norm. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M9">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M10">View MathML</a> be a continuous function verifying the following conditions:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M12">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M14">View MathML</a>;

3. f is locally Lipschitz in the second variable.

A solution of (1.1) is a function x with values in the ambient manifoldM, defined on an open real interval J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M15">View MathML</a>, bounded and uniformly continuous on any closed half-line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M16">View MathML</a> such that the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M17">View MathML</a> is eventually verified. We use here the standard notation in functional equations: whenever it makes sense, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M18">View MathML</a> denotes the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M19">View MathML</a>.

To proceed with the exposition of our problem, we need some further notation. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21">View MathML</a> denotes the constant p-valued function defined on ℝ or on any convenient subinterval of ℝ. The actual domain of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21">View MathML</a> will be clear from the context. Moreover, given any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M24">View MathML</a> stands for the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M25">View MathML</a>. All the functions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M24">View MathML</a> will be considered defined on the same interval, suggested by the context. By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27">View MathML</a> we mean the set of all continuous T-periodic maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28">View MathML</a>. This set, which contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M29">View MathML</a>, is a metric subspace of the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30">View MathML</a> with the standard supremum norm. We call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M31">View MathML</a> a T-periodic pair of equation (1.1) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28">View MathML</a> is a solution of (1.1) corresponding to λ. Among these pairs, we distinguish the trivial ones, that is, the elements of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M33">View MathML</a>, which can be isometrically identified with M. Notice that any T-periodic pair of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34">View MathML</a> is trivial since the function x turns out to be necessarily constant. An element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a> will be called a bifurcation point of (1.1) if any neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M36">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M37">View MathML</a> contains nontrivial T-periodic pairs. Roughly speaking, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a> is a bifurcation point if any of its neighborhoods in M contains T-periodic orbits corresponding to arbitrarily small values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M39">View MathML</a>.

The main outcome of this paper, Theorem 3.3 below, is a global continuation result for T-periodic solutions of equation (1.1). That is, given an open subset Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M40">View MathML</a>, it is a result which provides sufficient conditions for the existence of a global bifurcating branch in Ω, meaning a connected subset of Ω of nontrivialT-periodic pairs whose closure in Ω is noncompact and intersects the set of trivialT-periodic pairs. The proof of Theorem 3.3 is based on a relation, obtained in a technical result, Lemma 3.8 below, between the degree (in an open subset of M) of the tangent vector field

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M41">View MathML</a>

and the fixed point index of a sort of Poincaré T-translation operator acting inside the Banach space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M42">View MathML</a>.

The prelude of our approach can be found in some papers of the last two authors (see, for instance, [1]), where the notions of degree of a tangent vector field and of fixed point index of a suitable Poincaré T-translation operator are related in order to get continuation results for ODEs on differentiable manifolds.

Theorem 3.3 extends and unifies two results recently obtained by the authors in [2] and [3]. In [2] the ambient manifold M is not necessarily compact, but our investigation regards delay differential equations with finite time lag. On the other hand, in [3] we consider RFDEs with infinite delay; nevertheless, in this case M is compact and the map f is defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M43">View MathML</a> with a topology which is too weak, making the continuity assumption on f a too heavy condition.

We point out that, in order to obtain our continuation result for RFDEs with infinite delay without assuming the compactness of the ambient manifold M, we had to tackle strong technical difficulties. Therefore, we were forced to undertake a thorough preliminary investigation on the general properties of RFDEs with infinite delay on (possibly) noncompact manifolds. This was the purpose of our recent paper [4].

In our opinion the existence of a global bifurcating branch ensured by Theorem 3.3 should hold also without the assumption that f is locally Lipschitz in the second variable. However, we are not able to prove or disprove this conjecture because of some difficulties arising in this case. One is that the uniqueness of the initial value problem for equation (1.1) is not ensured and, consequently, a Poincaré T-translation operator is not defined as a single valued map. A classical tool to overcome this obstacle, usually applied in analogous problems, consists in considering a sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44">View MathML</a> maps approximating f. In our situation, however, because of the peculiar domain of f, we do not know how to realize this approach, and this is another difficulty.

We conclude the paper with some consequences of Theorem 3.3. One is a Rabinowitz-type global bifurcation result [5] obtained by assuming that the degree of the above tangent vector field w is nonzero on an open subset of M. Another corollary is deduced when M is compact: we get an existence result already proved in [6], and we extend an analogous one obtained in [3] in which the continuity assumption on f is too heavy. A third interesting case occurs when the degree of w is nonzero on a relatively compact open subset of M and suitable a priori bounds hold for the T-periodic orbits of equation (1.1): in this case, we obtain a continuation principle à la Mawhin [7,8].

The different and related cases of RFDEs with finite delay in Euclidean spaces have been investigated by many authors. For general reference, we suggest the monograph by Hale and Verduyn Lunel [9]. We refer also to the works of Gaines and Mawhin [10], Nussbaum [11,12] and Mallet-Paret, Nussbaum and Paraskevopoulos [13]. For RFDEs with infinite delay in Euclidean spaces, we recommend the article of Hale and Kato [14], the book by Hino, Murakami and Naito [15], and the more recent paper of Oliva and Rocha [16]. For RFDEs with finite delay on manifolds, we suggest the papers of Oliva [17,18]. Finally, for RFDEs with infinite delay on manifolds we cite [4].

2 Preliminaries

2.1 Fixed point index

We recall that a metrizable space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a> is an absolute neighborhood retract (ANR) if, whenever it is homeomorphically embedded as a closed subset C of a metric space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46">View MathML</a>, there exist an open neighborhood V of C in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46">View MathML</a> and a retraction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M48">View MathML</a> (see, e.g., [19,20]). Polyhedra and differentiable manifolds are examples of ANRs. Let us also recall that a continuous map between topological spaces is called locally compact if each point in its domain has a neighborhood whose image is contained in a compact set.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a> be a metric ANR and consider a locally compact (continuous) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>-valued map k defined on a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M51">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>. Given an open subset U of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a> contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M51">View MathML</a>, if the set of fixed points of k in U is compact, the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M55">View MathML</a> is called admissible. We point out that such a condition is clearly satisfied if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M57">View MathML</a> is compact and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M58">View MathML</a> for all p in the boundary of U. To any admissible pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M55">View MathML</a>, one can associate an integer <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M60">View MathML</a> - the fixed point index of k in U - which satisfies properties analogous to those of the classical Leray-Schauder degree [21]. The reader can see, for instance, [12,22-24] for a comprehensive presentation of the index theory for ANRs. As regards the connection with the homology theory, we refer to standard algebraic topology textbooks (e.g., [25,26]).

We summarize below the main properties of the fixed point index.

• (Existence) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M61">View MathML</a>, thenkadmits at least one fixed point inU.

• (Normalization) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>is compact, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M63">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M64">View MathML</a>denotes the Lefschetz number ofk.

• (Additivity) Given two disjoint open subsets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M65">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M66">View MathML</a>ofU, if any fixed point ofkinUis contained in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M67">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M68">View MathML</a>.

• (Excision) Given an open subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M65">View MathML</a>ofU, ifkhas no fixed points in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M70">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M71">View MathML</a>.

• (Commutativity) Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46">View MathML</a>be metric ANRs. Suppose thatUandVare open subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M46">View MathML</a>respectively and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M76">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M77">View MathML</a>are locally compact maps. Assume that the set of fixed points of eitherhkin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M78">View MathML</a>orkhin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M79">View MathML</a>is compact. Then the other set is compact as well and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M80">View MathML</a>.

• (Generalized homotopy invariance) LetIbe a compact real interval andWbe an open subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M81">View MathML</a>. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M82">View MathML</a>, denote<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M83">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M84">View MathML</a>be a locally compact map such that the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M85">View MathML</a>is compact. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M86">View MathML</a>is independent ofλ.

2.2 Degree of a vector field

Let us recall some basic notions on degree theory for tangent vector fields on differentiable manifolds. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M87">View MathML</a> be a continuous (autonomous) tangent vector field on a smooth manifold M, and let U be an open subset of M. We say that the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88">View MathML</a> is admissible (or, equivalently, that v is admissible in U) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M89">View MathML</a> is compact. In this case, one can assign to the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88">View MathML</a> an integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91">View MathML</a>, called the degree (or Euler characteristic, or rotation) of the tangent vector field v in U which, roughly speaking, counts algebraically the number of zeros of v in U (for general references, see, e.g., [27-30]). Notice that the condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M89">View MathML</a> to be compact is clearly satisfied if U is a relatively compact open subset of M and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M93">View MathML</a> for all p in the boundary of U.

As a consequence of the Poincaré-Hopf theorem, when M is compact, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M94">View MathML</a> equals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M95">View MathML</a>, the Euler-Poincaré characteristic of M.

In the particular case when U is an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M97">View MathML</a> is just the classical Brouwer degree of v in U when the map v is regarded as a vector field; namely, the degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M98">View MathML</a> of v in U with target value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M99">View MathML</a>. All the standard properties of the Brouwer degree in the flat case, such as homotopy invariance, excision, additivity, existence, still hold in the more general context of differentiable manifolds. To see this, one can use an equivalent definition of degree of a tangent vector field based on the fixed point index theory as presented in [1] and [31].

Let us stress that, actually, in [1] and [31] the definition of degree of a tangent vector field on M is given in terms of the fixed point index of a Poincaré-type translation operator associated to a suitable ODE on M. Such a definition provides a formula that will play a central role in Lemma 3.8 below, and this will be a crucial step in the proof of our main result.

We point out that no orientability of M is required for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91">View MathML</a> to be defined. This highlights the fact that the extension of the Brouwer degree for tangent vector fields in the non-flat case does not coincide with the one regarding maps between oriented manifolds with a given target value (as illustrated, for example, in [28,29]). This dichotomy of the notion of degree in the non-flat situation is not evident in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>: it is masked by the fact that an equation of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M102">View MathML</a> can be written as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M103">View MathML</a>. Anyhow, in the context of RFDEs (ODEs included), it is the degree of a vector field that plays a significative role.

It is known that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M88">View MathML</a> is admissible, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M105">View MathML</a>

(2.1)

where m denotes the dimension of M. Moreover, if v has an isolated zero p and U is an isolating (open) neighborhood of p, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M91">View MathML</a> is called the index of v at p. The excision property ensures that this is a well-defined integer.

2.3 Retarded functional differential equations

Given an arbitrary subset A of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M108">View MathML</a> the set of bounded and uniformly continuous maps from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109">View MathML</a> into A. For brevity, we will use the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M110">View MathML</a>

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M111">View MathML</a> is a Banach space, being closed in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M112">View MathML</a> of the bounded and continuous functions from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a> (endowed with the standard supremum norm).

Throughout the paper, the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a> will be denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M116">View MathML</a> and the norm in the infinite dimensional space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M111">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M118">View MathML</a>. Thus, the distance between two elements ϕ and ψ of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119">View MathML</a> will be denoted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M120">View MathML</a>, even when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M121">View MathML</a> does not belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119">View MathML</a>. We observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M119">View MathML</a>, as a metric space, is complete if and only if A is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>.

Let M be a boundaryless smooth manifold in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>. A continuous map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M126">View MathML</a>

is said to be a retarded functional tangent vector field overM if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M127">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M128">View MathML</a>. In the sequel, any map with this property will be briefly called a functional field (overM).

Let us consider a retarded functional differential equation (RFDE) of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M129">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M130">View MathML</a> is a functional field over M. Here, as usual and whenever it makes sense, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M131">View MathML</a>, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M132">View MathML</a> we mean the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M19">View MathML</a>.

A solution of (2.2) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134">View MathML</a>, defined on an open real interval J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M135">View MathML</a>, bounded and uniformly continuous on any closed half-line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M136">View MathML</a>, which verifies eventually the equality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134">View MathML</a> is a solution of (2.2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M139">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M140">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M141">View MathML</a> such that x is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44">View MathML</a> on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M143">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M145">View MathML</a>. Observe that the derivative of a solution x may not exist at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M146">View MathML</a>. However, the right derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M147">View MathML</a> of x at τ always exists and is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M148">View MathML</a>. Also, notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M149">View MathML</a> is a continuous curve in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150">View MathML</a> since x is uniformly continuous on any closed half-line <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M151">View MathML</a> of J.

A solution of (2.2) is said to be maximal if it is not a proper restriction of another solution. As in the case of ODEs, Zorn’s lemma implies that any solution is the restriction of a maximal solution.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M152">View MathML</a>, let us associate to equation (2.2) the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M153">View MathML</a>

(2.3)

A solution of (2.3) is a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M134">View MathML</a> of (2.2) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M137">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M157">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M158">View MathML</a>.

The continuous dependence of the solutions on initial data is stated in Theorem 2.1 below and is a straightforward consequence of Theorem 4.4 of [4].

Theorem 2.1LetMbe a boundaryless smooth manifold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M159">View MathML</a>be a functional field. Assume, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M160">View MathML</a>, the uniqueness of the maximal solution of problem (2.3). Then, given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161">View MathML</a>, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M162">View MathML</a>

is open and the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M163">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M164">View MathML</a>is the unique maximal solution of problem (2.3), is continuous.

More generally, we will need to consider initial value problems depending on a parameter such as equation (1.1) with the initial condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M158">View MathML</a>. For these problems the continuous dependence is ensured by the following consequence of Theorem 2.1.

Corollary 2.2 (Continuous dependence)

LetMbe a boundaryless smooth manifold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M166">View MathML</a>a parametrized functional field. For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M167">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M152">View MathML</a>, assume the uniqueness of the maximal solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M169">View MathML</a>

(2.4)

Then, given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161">View MathML</a>, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M171">View MathML</a>

is open and the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M172">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M173">View MathML</a>is the unique maximal solution of problem (2.4), is continuous.

Proof

Apply Theorem 2.1 to the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M174">View MathML</a>

that can be regarded as an initial value problem of a RFDE on the ambient manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M175">View MathML</a>. □

In Theorem 2.1 and in Corollary 2.2 above, the hypothesis of the uniqueness of the maximal solution of problems (2.3) and (2.4) is essential in order to make their statements meaningful. Sufficient conditions for the uniqueness are presented in Remark 2.3 below.

Remark 2.3 A functional field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M130">View MathML</a> is said to be compactly Lipschitz (for short, c-Lipschitz) if, given any compact subset Q of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M179">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M180">View MathML</a>. Moreover, we will say that g is locally c-Lipschitz if for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M181">View MathML</a> there exists an open neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M182">View MathML</a> in which g is c-Lipschitz. In spite of the fact that a locally Lipschitz map is not necessarily (globally) Lipschitz, one could actually show that if g is locally c-Lipschitz, then it is also (globally) c-Lipschitz. As a consequence, if g is locally Lipschitz in the second variable, then it is c-Lipschitz as well. In [4] we proved that if g is a c-Lipschitz functional field, then problem (2.3) has a unique maximal solution for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M160">View MathML</a>. For a characterization of compact subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150">View MathML</a> see, e.g., [[32], Part 1, IV.6.5].

We close this section with the following lemma whose elementary proof is given for the sake of completeness.

Lemma 2.4Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M185">View MathML</a>be a continuous map between metric spaces and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M186">View MathML</a>be a sequence of continuous functions from a compact interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M187">View MathML</a> (or, more generally, from a compact space) into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M189">View MathML</a>converges to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M190">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M191">View MathML</a>, then also<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M192">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M191">View MathML</a>.

Proof Notice that if K is a compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M45">View MathML</a>, then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M195">View MathML</a> there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M196">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M197">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M198">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M199">View MathML</a> imply <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M200">View MathML</a>. Now, our assertion follows immediately by taking the compact K to be the image of the limit function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M201">View MathML</a>. □

3 Branches of periodic solutions

Let M be a boundaryless smooth m-dimensional manifold in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M204">View MathML</a>

denote the metric subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M42">View MathML</a> of the M-valued continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M207">View MathML</a>

Moreover, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30">View MathML</a> the Banach space of the continuous T-periodic maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M209">View MathML</a> (with the standard supremum norm) and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27">View MathML</a> the metric subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M30">View MathML</a> of the M-valued maps. Observe that, since M is locally compact, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27">View MathML</a> (but not <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150">View MathML</a>) are locally complete. Moreover, they are complete if and only if M is closed.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M215">View MathML</a> be a functional field over M. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M161">View MathML</a>, assume that f is T-periodic in the first variable. Consider the following RFDE depending on a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M218">View MathML</a>

(3.1)

As in the introduction, we call <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M31">View MathML</a> a T-periodic pair (of (3.1)) if the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M28">View MathML</a> is a (T-periodic) solution of (3.1) corresponding to λ. Let us denote by X the set of all T-periodic pairs of (3.1). Lemma 3.1 below states some properties of X that will be used in the sequel.

Lemma 3.1The setXis closed in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a>and locally compact.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M222">View MathML</a> be a sequence of T-periodic pairs of (3.1) converging to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M223">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a>. Because of Lemma 2.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M225">View MathML</a> converges uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M226">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M228">View MathML</a> uniformly and, therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M229">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M230">View MathML</a> belongs to X. This proves that X is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a>.

Now, as observed above, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27">View MathML</a> is locally complete. Consequently, X is locally complete as well, as a closed subset of a locally complete space. Moreover, by using Ascoli’s theorem, we get that it is actually a locally compact space. □

We recall that, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a>, with the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M21">View MathML</a> we mean the constant p-valued function defined on some real interval that will be clear from the context. Moreover, a T-periodic pair of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235">View MathML</a> is said to be trivial, and an element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a> is a bifurcation point of equation (3.1) if any neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M238">View MathML</a> contains a nontrivial T-periodic pair (i.e., a T-periodic pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M239">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a>). In some sense, p is a bifurcation point if, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a> sufficiently small, there are T-periodic orbits of (3.1) arbitrarily close to p.

In the sequel, we are interested in the existence of branches of nontrivial T-periodic pairs that, roughly speaking, emanate from a trivial pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M242">View MathML</a>, with p a bifurcation point of (3.1). To this end, we introduce the mean value tangent vector field <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M243">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M244">View MathML</a>

(3.2)

Throughout the paper, w will play a crucial role in obtaining our continuation results for (3.1). First, in Theorem 3.2 below, we provide a necessary condition for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M4">View MathML</a> to be a bifurcation point.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M246">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34">View MathML</a>is an accumulation point of nontrivialT-periodic pairs of (3.1). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M248">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M249">View MathML</a>, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M251">View MathML</a>. Thus, any bifurcation point of (3.1) is a zero ofw.

Proof By assumption there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M252">View MathML</a> of T-periodic pairs of (3.1) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M255">View MathML</a> uniformly on ℝ. As proved in Lemma 3.1, the set X of the T-periodic pairs is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a>. Thus, the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M34">View MathML</a> belongs to X and, consequently, the function x must be constant, say <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M258">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M259">View MathML</a>. Clearly, the point p is a bifurcation point of (3.1).

Now, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M260">View MathML</a>, recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M261">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M262">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M263">View MathML</a>

Observe that the sequence of curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M264">View MathML</a> converges uniformly to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M265">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266">View MathML</a>. Hence, because of Lemma 2.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M267">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266">View MathML</a> and the assertion follows passing to the limit in the above integral. □

Let now Ω be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M238">View MathML</a>. Our main result (Theorem 3.3 below) provides a sufficient condition for the existence of a bifurcation point p in M with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M270">View MathML</a>. More precisely, we give conditions which ensure the existence of a connected subset of Ω of nontrivial T-periodic pairs of equation (3.1) (a global bifurcating branch for short), whose closure in Ω is noncompact and intersects the set of trivial T-periodic pairs contained in Ω.

Theorem 3.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M271">View MathML</a>be a boundaryless smooth manifold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M215">View MathML</a>be a functional field onM, T-periodic in the first variable and locally Lipschitz in the second one, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M273">View MathML</a>be the autonomous tangent vector field

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M274">View MathML</a>

Let Ω be an open subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M276">View MathML</a>be the map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M277">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M278">View MathML</a>is defined and nonzero. Then there exists a connected subset of Ω of nontrivialT-periodic pairs of equation (3.1) whose closure in Ω is noncompact and intersects<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M279">View MathML</a>in a (nonempty) subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M280">View MathML</a>.

Remark 3.4 (On the meaning of global bifurcating branch)

In addition to the hypotheses of Theorem 3.3, assume that f sends bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a> into bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>, and that M is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a> (or, more generally, that the closure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284">View MathML</a> of Ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M221">View MathML</a> is complete).

Then a connected subset Γ of Ω as in Theorem 3.3 is either unbounded or, if bounded, its closure<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284">View MathML</a>reaches the boundaryΩ of Ω.

To see this, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> is bounded. Then, being <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M289">View MathML</a> bounded, because of Ascoli’s theorem, Γ is actually totally bounded. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> is compact, being totally bounded and, additionally, complete since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M291">View MathML</a> is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M292">View MathML</a>. On the other hand, according to Theorem 3.3, the closure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M293">View MathML</a> of Γ in Ω is noncompact. Consequently, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M294">View MathML</a> is nonempty, and this means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> reaches the boundary of Ω.

The proof of Theorem 3.3 requires some preliminary steps. In the first one, we define a parametrized Poincaré-type T-translation operator whose fixed points are the restrictions to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a> of the T-periodic solutions of (3.1). For this purpose, we need to introduce a suitable backward extension of the elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>. The properties of such an extension are contained in Lemma 3.5 below, obtained in [33]. In what follows, by a T-periodic map on an interval J, we mean the restriction to J of a T-periodic map defined on ℝ.

Lemma 3.5There exist an open neighborhoodUof<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M298">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>and a continuous map fromUto<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M301">View MathML</a>, with the following properties:

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302">View MathML</a>is an extension ofψ;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302">View MathML</a>isT-periodic on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M304">View MathML</a>;

3. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302">View MathML</a>isT-periodic on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M109">View MathML</a>, whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M307">View MathML</a>.

Let now U be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a> as in the previous lemma and let f be as in Theorem 3.3. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M310">View MathML</a>, consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M311">View MathML</a>

(3.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302">View MathML</a> is the extension of ψ as in Lemma 3.5.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M313">View MathML</a>

The set D is nonempty since it contains <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M314">View MathML</a> (notice that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315">View MathML</a>, the solution of problem (3.3) is constant for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M157">View MathML</a>). Moreover, it follows by Corollary 2.2 that D is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M317">View MathML</a>.

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M318">View MathML</a>, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319">View MathML</a> the maximal solution of problem (3.3) and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M320">View MathML</a>

by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M321">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M322">View MathML</a> is the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M323">View MathML</a> to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a>.

The following lemmas regard crucial properties of the operator P. The proof of the first one is standard and will be omitted.

Lemma 3.6The fixed points of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325">View MathML</a>correspond to theT-periodic solutions of equation (3.1) in the following sense: ψis a fixed point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325">View MathML</a>if and only if it is the restriction to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a>of aT-periodic solution.

Lemma 3.7The operatorPis continuous and locally compact.

Proof The continuity of P follows immediately from the continuous dependence on data stated in Corollary 2.2 and by the continuity of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M301">View MathML</a> of Lemma 3.5 and of the map that associates to any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M329">View MathML</a> its restriction to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a>.

Let us prove that P is locally compact. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M331">View MathML</a> and denote, for simplicity, by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332">View MathML</a> the maximal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M333">View MathML</a> of (3.3) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M334">View MathML</a>. Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332">View MathML</a> is defined at least up to T and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M336">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M337">View MathML</a>. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M338">View MathML</a>

Observe that K is compact, being the image of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M339">View MathML</a> under the (continuous) curve <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M340">View MathML</a>. Let O be an open neighborhood of K in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M342">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M343">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M344">View MathML</a>. Let us show that there exists an open neighborhood W of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M345">View MathML</a> in D such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M346">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M347">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319">View MathML</a> is the maximal solution of (3.3) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M350">View MathML</a>. By contradiction, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M351">View MathML</a> suppose there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M352">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M353">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M354">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M355">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M356">View MathML</a> denotes the maximal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M357">View MathML</a> of (3.3) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M358">View MathML</a>. We may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M359">View MathML</a>. Now, from the fact that in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M150">View MathML</a> the convergence is uniform, we get the equicontinuity of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M361">View MathML</a>. This easily implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M362">View MathML</a>. A contradiction, since O is open and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M363">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M364">View MathML</a>. Thus, the existence of the required W is proved. Consequently, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M365">View MathML</a>, the maximal solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M319">View MathML</a> of (3.3) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M350">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M368">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M266">View MathML</a>.

Therefore, by Ascoli’s theorem and taking into account the local completeness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>, we get that P maps W into a compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>. This proves that P is locally compact. □

The following result establishes the relationship between the fixed point index of the Poincaré-type operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M325">View MathML</a> and the degree of the mean value vector field w. It will be crucial in the proof of Lemma 3.10.

Lemma 3.8Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373">View MathML</a>be an open subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M375">View MathML</a>is compact and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M376">View MathML</a>be such that

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M377">View MathML</a>is contained in the domainDofP;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M378">View MathML</a>is relatively compact;

(c) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M379">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M380">View MathML</a>andψin the boundary<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M381">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373">View MathML</a>.

Consider the open set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M383">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M384">View MathML</a>is well defined and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M385">View MathML</a>

Proof Let U be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a> as in Lemma 3.5. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M387">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M310">View MathML</a>, consider the initial value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M390">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M302">View MathML</a> is associated to ψ as in Lemma 3.5. Since f is locally Lipschitz in the second variable, then it is easy to see that w is locally Lipschitz as well. Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M392">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388">View MathML</a>, the uniqueness of the solution of problem (3.4) is ensured (recall Remark 2.3). Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M394">View MathML</a> the maximal solution of problem (3.4), and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M395">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M396">View MathML</a>

Corollary 2.2 implies that E is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M397">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398">View MathML</a> is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M399">View MathML</a> because of the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M400">View MathML</a>. Moreover, observe that the slice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M401">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315">View MathML</a> coincides with U and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398">View MathML</a> is contained in the domain D of the operator P defined above. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M405">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M406">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M407">View MathML</a> coincides with P on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398">View MathML</a>, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M409">View MathML</a> is the (infinite dimensional) operator associated to the undelayed problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M410">View MathML</a>

As in Lemmas 3.6 and 3.7, one can show that the fixed points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M411">View MathML</a> correspond to the T-periodic solutions of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M412">View MathML</a>

and that H is continuous and locally compact.

The assertion now will follow by proving some intermediate results on the homotopy H. These results will be carried out in several steps. In what follows set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M413">View MathML</a>

and, according to our notation,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M414">View MathML</a>

Step 1. There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M415">View MathML</a>and an open subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>, containing<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M418">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M419">View MathML</a>, and such that

(a′) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M420">View MathML</a>(i.e., for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M421">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M422">View MathML</a>is defined in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M423">View MathML</a>);

(b′) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M424">View MathML</a>is relatively compact.

To prove Step 1, observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M425">View MathML</a> is compact and contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M426">View MathML</a>, which is open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M427">View MathML</a>, and recall that H is locally compact.

Step 2. For small values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M429','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M429">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M430">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388">View MathML</a>.

By contradiction, suppose there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M432">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M426">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M436">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M437','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M437">View MathML</a>. Without loss of generality, taking into account (b′), we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438">View MathML</a> and also that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M439">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M440">View MathML</a> the T-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M441">View MathML</a> of (3.4) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M442">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M443">View MathML</a> is the restriction of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M356">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M446">View MathML</a> converges uniformly on ℝ to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M447">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M332">View MathML</a> is the solution of (3.4) corresponding to the fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M449">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M450">View MathML</a>. Therefore, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M248">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M452','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M452">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M227">View MathML</a> and, as in the proof of Theorem 3.2, we can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M251">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M455">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M456">View MathML</a>, contradicting the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416">View MathML</a>. This proves Step 2.

Step 3. For small values of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a>,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M459">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M460">View MathML</a>.

The proof is analogous to that of Step 2, noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M461">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M462">View MathML</a> and taking into account assumption b) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M463','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M463">View MathML</a> is closed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>.

Step 4. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M465">View MathML</a>be defined by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M466">View MathML</a>and consider the open set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M467">View MathML</a>. Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M468">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M469">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M470">View MathML</a>.

By contradiction, suppose there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M471">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M398">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M475','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M475">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M476">View MathML</a>. Without loss of generality, taking into account (b′), we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438">View MathML</a>. Therefore, by the continuity of H, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M478">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M449">View MathML</a> is a constant function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M480">View MathML</a>. This is impossible, since any constant function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416">View MathML</a> is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482">View MathML</a>.

Step 5. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M483">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M484">View MathML</a>be as in Step 4 and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M485','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M485">View MathML</a>be theT-translation operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M486','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M486">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M487">View MathML</a>is the maximal solution of the undelayed problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M488">View MathML</a>

Then, for small values ofλ,<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M489">View MathML</a>is defined and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M490">View MathML</a>

To see this, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M465','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M465">View MathML</a> be as in Step 4 and, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M492','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M492">View MathML</a>, define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M493">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M494">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M495">View MathML</a>. Clearly, k is a locally compact map since it takes values in the locally compact space M. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M496">View MathML</a> is actually compact since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M497','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M497">View MathML</a> is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M424">View MathML</a> which is relatively compact by (b′) of Step 1. Now, observe that the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M499">View MathML</a> coincides with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M500">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482">View MathML</a> and that the set of fixed points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M500">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M503','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M503">View MathML</a> is compact by (b′) of Step 1 and is contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482">View MathML</a> by Step 4. Thus, the set of fixed points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M499','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M499">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M482">View MathML</a> is compact so that, by applying the commutativity property of the fixed point index to the maps k and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M496">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M508','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M508">View MathML</a>

Consequently, since it is easy to verify that the composition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M509','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M509">View MathML</a> coincides with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M510','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M510">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M511','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M511">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M512','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M512">View MathML</a>

and, because of Step 4, by the excision property of the index,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M513','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M513">View MathML</a>

To complete the proof of Step 5, let us show that for λ sufficiently small, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M514','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M514">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M515','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M515">View MathML</a>. By contradiction, suppose there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M516','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M516">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M517','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M517">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M434">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M519','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M519">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M520','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M520">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M521','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M521">View MathML</a>. Hence, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M522','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M522">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M523','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M523">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M524','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M524">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M525','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M525">View MathML</a>. Because of (b′) of Step 1, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M438">View MathML</a> so that, in particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M527','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M527">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M528','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M528">View MathML</a>. Now, by an argument similar to that used in the proof of Theorem 3.2, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M529','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M529">View MathML</a> is constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M530','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M530">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M531','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M531">View MathML</a>. Moreover, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M532','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M532">View MathML</a>, we also obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M533','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M533">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M534','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M534">View MathML</a>, contradicting the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M483">View MathML</a>. Finally, again by excision, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M536','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M536">View MathML</a>

and thus Step 5 is proved.

Let us now go back to the proof of our lemma. Step 1 and Step 2 above imply that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M537','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M537">View MathML</a> and an open subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a>, containing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M418">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M419">View MathML</a> and such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M542','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M542">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M543','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M543">View MathML</a> is defined and is independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M388">View MathML</a>. Moreover, reducing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M545','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M545">View MathML</a> if necessary, by Step 3 and by assumption (b), it follows that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M546','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M546">View MathML</a>, the fixed points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M547','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M547">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373">View MathML</a> are a compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M416">View MathML</a>. Therefore, by the excision property and the homotopy invariance of the index, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M550','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M550">View MathML</a>

On the other hand, by Step 5, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a> is sufficiently small, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M552','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M552">View MathML</a>

Moreover, as shown in [1],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M553','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M553">View MathML</a>

Finally, notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M384">View MathML</a> is well defined since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M555','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M555">View MathML</a> is compact being homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M556','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M556">View MathML</a>. Also, observe that there are no zeros of w in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M557','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M557">View MathML</a>. Thus, by the excision property of the degree, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M558','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M558">View MathML</a>

This shows that for small values of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M560','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M560">View MathML</a>. The assertion of the lemma now follows by applying the homotopy invariance of the fixed point index to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M561','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M561">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373">View MathML</a>. □

Lemma 3.10 below, whose proof makes use of the following Wyburn-type topological lemma, is another important step in the construction of the proof of Theorem 3.3.

Lemma 3.9 ([31])

LetKbe a compact subset of a locally compact metric spaceY. Assume that any compact subset ofYcontainingKhas nonempty boundary. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M563','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M563">View MathML</a>contains a connected set whose closure is noncompact and intersectsK.

Before presenting Lemma 3.10, we introduce the sets

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M564','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M564">View MathML</a>

and we recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M565','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M565">View MathML</a> denotes the set of zeros of the tangent vector field w.

Lemma 3.10LetYbe a locally compact open subset of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M566">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M567','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M567">View MathML</a>is compact and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M568','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M568">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M569','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M569">View MathML</a>, is an isolating neighborhood of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M570">View MathML</a>. Then the pair<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571">View MathML</a>verifies the assumptions of Lemma 3.9.

Proof First of all, observe that by Lemma 3.7, S is closed in D and locally compact. In addition, K is clearly nonempty being <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572">View MathML</a>. Now, let G be an open subset of D such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M573','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M573">View MathML</a>

To prove the assertion, suppose by contradiction that there exists a compact open neighborhood C of K in Y. Consequently, we can find an open subset W of G such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M574','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M574">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M575','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M575">View MathML</a>. Therefore, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M576','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M576">View MathML</a> the slice

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M577','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M577">View MathML</a>

we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M578','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M578">View MathML</a> is a compact subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M212">View MathML</a> and is contained in the open slice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M580','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M580">View MathML</a> of W at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M315">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M373">View MathML</a> be an open subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M583','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M583">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M584','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M584">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M585','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M585">View MathML</a>. Since C is compact and because of the local compactness of P, we may suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M586">View MathML</a> is relatively compact. Consequently, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M376">View MathML</a> such that

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M588','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M588">View MathML</a>;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M589','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M589">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M590','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M590">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M591','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M591">View MathML</a> (here, as usual, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M592','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M592">View MathML</a> denotes the slice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M593','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M593">View MathML</a>).

Notice that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M594','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M594">View MathML</a> is relatively compact. This follows easily from the above condition 1 and the relative compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M586','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M586">View MathML</a>.

We can now apply Lemma 3.8 and the excision properties of the fixed point index and of the degree obtaining, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M596','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M596">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M597','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M597">View MathML</a>

(3.5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M383">View MathML</a>. Observe that V is an isolating neighborhood of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M570','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M570">View MathML</a>. Thus, by formula (2.1), by the above equalities (3.5) and the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M601','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M601">View MathML</a>

Since C is compact, by the generalized homotopy invariance property of the fixed point index, we get that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M602','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M602">View MathML</a> does not depend on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M240">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M604','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M604">View MathML</a>

On the other hand, because of the compactness of C, for some positive <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M605','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M605">View MathML</a> the slice <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M606','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M606">View MathML</a> is empty. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M607','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M607">View MathML</a>

and we have a contradiction. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571">View MathML</a> verifies the assumptions of Lemma 3.9 and the proof is complete. □

Proof of Theorem 3.3 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M609','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M609">View MathML</a> be the isometry given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M610','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M610">View MathML</a>, where ψ is the restriction of x to the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M206">View MathML</a>. As previously, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M612','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M612">View MathML</a> denote the set of the T-periodic pairs of (3.1) and, as in Lemma 3.10, let S be the set of the pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M613','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M613">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M614','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M614">View MathML</a>. Observe that S is actually contained in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M615','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M615">View MathML</a>. Taking into account Lemma 3.6, X and S correspond under ρ. Analogously to the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M616','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M616">View MathML</a>, let us denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M617','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M617">View MathML</a>

In addition, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M618','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M618">View MathML</a>

Theorem 3.2 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M619','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M619">View MathML</a> is a closed subset of X. Therefore, it is locally compact since so is X according to Lemma 3.1. Now, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M620','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M620">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621">View MathML</a> is locally compact, being open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M622','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M622">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M623','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M623">View MathML</a>

is locally compact and open in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M566','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M566">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M625">View MathML</a> and K the subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621">View MathML</a> and Y defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M627','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M627">View MathML</a>

Now, observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M628','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M628">View MathML</a> is an isolating neighborhood of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M629','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M629">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M630','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M630">View MathML</a>, we can apply Lemma 3.10 concluding that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571">View MathML</a> verifies the assumptions of Lemma 3.9. Therefore, also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M632">View MathML</a> verifies the same assumptions since the pairs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M571">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M632','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M632">View MathML</a> correspond under the isometry ρ. Therefore, Lemma 3.9 implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M635','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M635">View MathML</a> contains a connected set Γ whose closure (in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621">View MathML</a>) is noncompact and intersects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M625','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M625">View MathML</a>. Now, observe that according to Theorem 3.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621">View MathML</a> is closed in Ω. Thus, the closures of Γ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M621">View MathML</a> and in Ω coincide. This concludes the proof. □

We give now some consequences of Theorem 3.3. The first one is in the spirit of a celebrated result due to Rabinowitz [5].

Corollary 3.11 (Rabinowitz-type global bifurcation result)

LetMandfbe as in Theorem 3.3. Assume thatMis closed in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>and thatfsends bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a>into bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>. LetVbe an open subset ofMsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M572">View MathML</a>, wherewis the mean value tangent vector field defined in formula (3.2). Then equation (3.1) has a connected subset of nontrivialT-periodic pairs whose closure contains some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M235">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M645','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M645">View MathML</a>, and is either unbounded or goes back to some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M646">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M647">View MathML</a>.

Proof Let Ω be the open set obtained by removing from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M648','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M648">View MathML</a> the closed set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M649','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M649">View MathML</a>. In other words,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M650','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M650">View MathML</a>

Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284">View MathML</a> is complete due to the closedness of M. Consider, by Theorem 3.3, a connected set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M652','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M652">View MathML</a> of nontrivial T-periodic pairs with noncompact closure (in Ω) and intersecting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M279">View MathML</a> in a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M654">View MathML</a>. Suppose that Γ is bounded. From Remark 3.4 it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M294">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M656','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M656">View MathML</a> denotes the closure of Γ in Ω, is nonempty and hence contains a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M646','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M646">View MathML</a> which does not belong to Ω, that is, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M647','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M647">View MathML</a>. □

Remark 3.12 The assumption of Corollary 3.11 above on the existence of an open subset V of M such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M659">View MathML</a> is clearly satisfied in the case when w has an isolated zero with nonzero index. For example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M660','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M660">View MathML</a> and w is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M44">View MathML</a> with injective derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M662','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M662">View MathML</a>, then p is an isolated zero of w and its index is either 1 or −1. In fact, in this case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M663','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M663">View MathML</a> sends <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M664','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M664">View MathML</a> into itself and, consequently, its determinant is well defined and nonzero. The index of p is just the sign of this determinant (see, e.g., [29]).

The next consequence of Theorem 3.3 provides an existence result for T-periodic solutions already obtained in [6]. Moreover, it improves an analogous result in [3], in which the map f is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M665','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M665">View MathML</a>, with the compact-open topology in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M666','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M666">View MathML</a>. In fact, such a coarse topology makes the assumption of the continuity of f a more restrictive condition than the one we require here.

Corollary 3.13LetMandfbe as in Theorem 3.3. Assume thatfsends bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a>into bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>. In addition, suppose thatMis compact with Euler-Poincaré characteristic<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M669','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M669">View MathML</a>. Then equation (3.1) has a connected unbounded set of nontrivialT-periodic pairs whose closure meets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M670">View MathML</a>. Therefore, since<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M27">View MathML</a>is bounded, equation (3.1) has aT-periodic solution for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M387">View MathML</a>.

Proof Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M673','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M673">View MathML</a>. By the Poincaré-Hopf theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M674','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M674">View MathML</a>

where w is the mean value tangent vector field defined in formula (3.2). The assertion follows from Corollary 3.11. □

Corollary 3.14 below is a kind of continuation principle in the spirit of a well-known result due to Jean Mawhin for ODEs in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>[7,8] and extends an analogous one for ODEs on differentiable manifolds [31]. In what follows, by a T-periodic orbit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M676','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M676">View MathML</a>, we mean the image of a T-periodic solution of this equation.

Corollary 3.14 (Mawhin-type continuation principle)

LetMandfbe as in Theorem 3.3 and letwbe the mean value tangent vector field defined in formula (3.2). Assume thatfsends bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a>into bounded subsets of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>. LetVbe a relatively compact open subset ofMand assume that

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M679','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M679">View MathML</a>along the boundary∂VofV;

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M659','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M659">View MathML</a>;

3. for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M681','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M681">View MathML</a>, theT-periodic orbits of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M682','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M682">View MathML</a>lying in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M683','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M683">View MathML</a>do not meet∂V.

Then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M684','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M684">View MathML</a>

has aT-periodic orbit in V.

Proof Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M685','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M685">View MathML</a>. Observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M686','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M686">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M687','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M687">View MathML</a>

According to Theorem 3.3, call Γ a connected subset of Ω of nontrivial T-periodic pairs of the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M17">View MathML</a>, whose closure in Ω is noncompact and intersects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M670','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M670">View MathML</a> in a subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M654','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M654">View MathML</a>.

As V has compact closure in M, then the closure of Ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M691','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M691">View MathML</a> is complete, being

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M692','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M692">View MathML</a>

Since f sends bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M177">View MathML</a> into bounded subsets of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M96">View MathML</a>, recalling Remark 3.4, one has that the closure <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> of Γ in the whole space (which coincides with the closure in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M284">View MathML</a>) must intersect Ω.

Now, because of the above condition 3, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> cannot contain elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M698','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M698">View MathML</a>. In addition, condition 1 and Theorem 3.2 imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M286">View MathML</a> does not contain elements of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M700','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M700">View MathML</a>. Therefore, the nonempty set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M701','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M701">View MathML</a> is composed of pairs of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M702','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M702">View MathML</a>, where x is a T-periodic solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M703','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/21/mathml/M703">View MathML</a> whose image is contained in V. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed to each part of the work equally. All authors read and approved the final version of the manuscript.

Acknowledgements

Dedicated to our friend and outstanding mathematician Jean Mawhin.

Pierluigi Benevieri is partially sponsored by Fapesp, Grant n. 2010/20727-4.

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