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

Open Access Research

A bifurcation problem for a class of periodically perturbed autonomous parabolic equations

Mikhail Kamenskii1, Boris Mikhaylenko1 and Paolo Nistri2*

Author affiliations

1 Department of Mathematics, Voronezh State University, Voronezh, 394006, Russia

2 Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche, Università di Siena, Siena, 53100, Italy

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Citation and License

Boundary Value Problems 2013, 2013:101  doi:10.1186/1687-2770-2013-101

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/101


Received:24 December 2012
Accepted:6 April 2013
Published:23 April 2013

© 2013 Kamenskii 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

The paper deals with the problem of the existence of a branch of T-periodic solutions originating from the isolated limit cycle of an autonomous parabolic equation in a Banach space when it is perturbed by a nonlinear T-periodic term of small amplitude.

We solve this problem by first introducing a novel integral operator, whose fixed points are T-periodic solutions of the considered equation and vice versa. Then we compute the Malkin bifurcation function associated to this integral operator and we provide conditions under which the well-known assumption of the existence of a simple zero of the Malkin bifurcation function guarantees the existence of the branch.

MSC: 35K58, 35B10, 35B20, 35B32.

Keywords:
autonomous parabolic equations; periodic perturbations; limit cycle; bifurcation; periodic solutions

1 Introduction

In recent years, bifurcation problems for smooth and nonsmooth dynamical systems have received a renewed attention and interest from different fields of engineering, physics and mathematics. We mention here, among others, the monographs [1-5] and the review papers [6,7]. Of particular interest is the study of the bifurcation of periodic solutions for periodically perturbed autonomous systems of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M1">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M3">View MathML</a>, ψ is T-periodic with respect to time and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> is a small parameter. Precisely, one seeks for the existence of a family of T-periodic solutions originating from a limit cycle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> of the autonomous unperturbed system.

Existence, uniqueness and asymptotic stability of bifurcating periodic solutions for system (1) are classical problems; see [8,9]. The main tool employed in these papers is the so-called Malkin bifurcation function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M6">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7">View MathML</a> is a T-periodic solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M8">View MathML</a>

the adjoint system of the linearized system

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M9">View MathML</a>

It is assumed that the linearized system has only one characteristic multiplier with absolute value 1.

Since the pioneering papers [8,9], a relevant bibliography devoted to this subject has been developed. From this bibliography, we quote in the sequel some of the papers more related to the present paper. In [10], the case when the cycle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> is not isolated was considered. By means of suitably defined bifurcation functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M12">View MathML</a>, called Melnikov subharmonic functions, the existence of periodic solutions near to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> was proved. The periods of the solutions are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M14">View MathML</a> ratio with respect to the period of the perturbation term. The case when 1 is not a simple multiplier of the linearized system was treated in [11]. The existence of at least two branches of T-periodic solutions originating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> is shown in [12,13] and their stability, in the sense of Lyapunov, follows from the results of [14,15]. Developments of the Malkin’s and Melnikov’s approaches have permitted to prove several results about the existence of bifurcating solutions in [16-20]. Furthermore, the use of a Melnikov function permits to detect chaotic behavior of a suitable iterate of the Poincaré map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M16">View MathML</a> associated to the differential equation (1), which is assumed to have a homoclinic orbit. Indeed, the existence of a simple zero of the considered Melnikov function ensures such a chaotic behavior; see [21,22].

Very recently, in [23], a new method to prove bifurcation of a branch of asymptotically stable periodic solutions to (1) has been proposed. The method consists first in converting the problem of finding fixed points of the singular Poincaré map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a>, associated to (1) into the problem of the existence of zeros of an equation of the form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M19">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M21">View MathML</a> are given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M23">View MathML</a> with singular <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M24">View MathML</a>. Then, by a convenient scaling of the variable x, we introduce an equivalent equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M25">View MathML</a>. For this equation, under the usual assumption of the existence of a simple zero of the Malkin bifurcation function associated to (1), the classical implicit function theorem can be applied to prove the existence of a branch of solutions originating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a>.

The same approach has been employed in [24] for a class of systems for which the resulting operators P and Q satisfy regularity conditions, which permit to apply the implicit function theorem, only along certain directions at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M27">View MathML</a> of the limit cycle. Conditions to ensure the existence of several branches of T-periodic solution emanating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> are provided by means of suitably defined Malkin bifurcation functions.

In all the papers cited before, the existence of periodic solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> small is a consequence of the application of a convenient version of the implicit function theorem. This requires, as assumed for system (1), that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M3">View MathML</a>. Under less restrictive regularity conditions, by using topological tools such as the coincidence degree [25], the Leray-Schauder degree and the related continuation principles [26-28], existence results for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> small have been proved in [29] when the autonomous system is Hamiltonian and in [30,31] when the limit cycle is isolated. More precisely, in [29,31] the existence of two branches of T-periodic solutions was proved. Roughly speaking, in these papers, the bifurcation functions are employed to guarantee that the topological degree of certain operators is different from zero, rather than for the application of an implicit function theorem.

Topological degree arguments have been also employed in [32] to show the existence of periodic solutions for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> small in the case when the unperturbed system is nonautonomous and the perturbation consists of two nonlinear periodic terms with multiplicative different powers of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a>. Finally, the behavior of the bifurcating periodic solutions when the perturbation vanishes has been studied in [33] for a nonsmooth system of the form (1) having an isolated limit cycle and in [34] for nonsmooth planar Hamiltonian systems.

A first attempt to extend to infinite dimensional bifurcation problem the approach outlined in [23] has been presented in [35], with the aim of studying the bifurcation of periodic solutions for a functional differential equation of neutral type. In the present paper, we precise and generalize the idea of how to use a suitable abstract Malkin bifurcation function to deal with infinite dimensional bifurcation problems. To this aim, we consider the following autonomous differential equation of parabolic type periodically perturbed by a nonlinear term of small amplitude:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M35">View MathML</a>

(2)

where A is the infinitesimal generator of a strongly continuous semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M37">View MathML</a>, acting in the Banach space E, satisfying the Radon-Nikodym property, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M38">View MathML</a> is twice continuously Frechét differentiable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M39">View MathML</a> is continuously Frechét differentiable with respect to x, ε and T-periodic with respect to time. The functions ϕ and ψ satisfy suitable condensivity conditions with respect to the Hausdorff measure of noncompactness. The crucial assumption is that the unperturbed equation at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40">View MathML</a> has a continuous T-periodic isolated solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M41">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M42">View MathML</a>.

The paper is organized as follows. In Section 2, we precise the conditions under which there is at least a branch of T-periodic solutions to (2) emanating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a>. This existence result follows from the application of [[36], Theorem 2]; this theorem relies on the method introduced in [23]. Precisely, to solve the bifurcation problem for (2), we introduce an equivalent integral equation whose zeros are the T-periodic solutions to (2) and that we rewrite in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M44">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M46">View MathML</a>. This equation has a branch of solutions originating from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M47">View MathML</a> if the Malkin bifurcation function given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M48">View MathML</a>

has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49">View MathML</a> as simple zero. Here, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M51">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53">View MathML</a> is the eigenvector corresponding to the simple eigenvalue 0 of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M55">View MathML</a> denotes the duality pairing of E with its dual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M56">View MathML</a>. The main difficulty to verify the conditions of [[36], Theorem 2] consists in proving that the zero eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57">View MathML</a> is simple. In fact, in this case, the assumption that the linearized equation, around the limit cycle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, of the autonomous system at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40">View MathML</a>, does not have neither T-periodic solutions linearly independent with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M61">View MathML</a> nor Floquet adjoint solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M61">View MathML</a> does not guarantee that the zero eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57">View MathML</a> is simple. To overcome this difficulty, we define in Section 3 a novel integral operator, equivalent to that associated to (2) with the property that for the resulting equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M64">View MathML</a>, the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a> has 0 as simple eigenvalue.

Furthermore, in Section 4, we calculate the Malkin bifurcation function associated to the integral operator introduced in Section 3, and we formulate in Theorem 2 the result of the existence of a branch of T-periodic solutions parameterized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> small. Proposition 1 of Section 5 states a somewhat surprising result: the Malkin functions associated to the two integral operators coincide and they have the common form of the classical Malkin function introduced for ordinary differential equations in finite dimensional spaces of the form (1). Finally, in Section 6, we provide a concrete example of a system of partial differential equations to which our abstract bifurcation result applies.

2 Assumptions and statement of the problem

The paper deals with the problem of the existence of bifurcation of T-periodic solutions for the T-periodically perturbed autonomous equation of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M67">View MathML</a>

(3)

from a T-periodic limit cycle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M5">View MathML</a> of the unperturbed system corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40">View MathML</a>. Here, A is the infinitesimal generator of a strongly continuous semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M71">View MathML</a>, acting in the Banach space E, which satisfies the Radon-Nikodym property; see [[37], Theorem 23, p.276]; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M73">View MathML</a> is T-periodic and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M74">View MathML</a>, the space of T-periodic continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M75">View MathML</a>.

Throughout the paper, we assume the following conditions on A, ϕ and ψ.

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M76">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M77">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M78">View MathML</a>

(H2) ϕ is twice continuously Fréchet differentiable, ψ is continuously Fréchet differentiable with respect to the pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M79">View MathML</a>. Moreover, for any nonempty, bounded set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M80">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M81">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M83">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M84">View MathML</a> is the Hausdorff measure of noncompactness [38].

(H3) The unperturbed equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M85">View MathML</a>

(4)

has a T-periodic isolated solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M42">View MathML</a>, hence the set of shifts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M87">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M88">View MathML</a>, represents a family of T-periodic solutions to (4). Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M90">View MathML</a> is a T-periodic solution to the linearized equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M91">View MathML</a>

(5)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M92">View MathML</a>. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M93">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a> and that (5) does not possess neither T-periodic solution linearly independent with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95">View MathML</a> nor Floquet adjoint solution to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, i.e., (5) does not have solutions of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M99">View MathML</a> is a T-periodic function.

We pose the following.

Problem To find conditions to ensure the existence of a branch of T-periodic solutions to (3) parameterized by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a>, originating, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49">View MathML</a>, from the family of T-periodic solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M58">View MathML</a>.

To solve this problem, we first reduce the existence of T-periodic solutions to (3) to the problem of finding fixed points of an integral equation. For this, we introduce the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M103">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M104">View MathML</a>

Therefore, if we let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M105">View MathML</a>

then a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106">View MathML</a> satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M107">View MathML</a>

(6)

is a solution to (3) and vice versa. Moreover, it is easy to verify that the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M108">View MathML</a>

is equivalent to the linearized unperturbed equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M109">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M110">View MathML</a>. Hence, we can rewrite (6) in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M111">View MathML</a>

(7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M45">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M46">View MathML</a> are defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M114">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M115">View MathML</a>

In conclusion, our problem will be solved if we show that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a> sufficiently small, equation (7) has a solution. To this end, it would be sufficient to verify the conditions of the following result.

Theorem 1 ([[36], Theorem 2])

LetBbe a Banach space, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M117">View MathML</a>be a twice continuously Fréchet differentiable map and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M118">View MathML</a>continuously Fréchet differentiable with respect to both the variables.

Assume that the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M119">View MathML</a>has one-dimensional set of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M120">View MathML</a>, parameterized by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, such that there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M122">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M124">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>. Assume that the eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M126">View MathML</a>is simple and the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M127">View MathML</a>is compact, whenever<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>. Consider the function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M129">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53">View MathML</a>is the eigenvector corresponding to the simple eigenvalue<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M131">View MathML</a>. Here, ∗ denotes the adjoint operator.

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M49">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M133">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M134">View MathML</a>the equation<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M64">View MathML</a>is solvable, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a>sufficiently small, in a neighborhood of the point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M137">View MathML</a>and the solution has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M138">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M139">View MathML</a>can be determined in explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].

As it has been observed in [39], the compactness of the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140">View MathML</a> can be replaced by the condensivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140">View MathML</a> with respect to the Hausdorff measure of noncompactness. Indeed, as it is shown in [40], under assumptions (H1)-(H2), the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M142">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140">View MathML</a>, see [[38], Theorem 1.5.4], are condensing with constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M144">View MathML</a>. Furthermore, [[38], Theorem 2.6.11] ensures that zero is an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a> of finite multiplicity. Therefore, under assumptions (H1)-(H3), one can easily verify that the conditions of the previous Theorem 1 are satisfied for (7) except the condition of the simplicity of the zero eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>. In fact, the assumption that (5) does not possess neither T-periodic solutions linearly independent with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95">View MathML</a>, nor Floquet adjoint solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95">View MathML</a>, whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, does not imply that the zero eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57">View MathML</a> is simple, despite the existence of a bijection between the T-periodic solutions to (5) and the T-periodic solutions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M152">View MathML</a>. Moreover, as it shown in [41] the simplicity of the zero eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M57">View MathML</a> does not imply that the T-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M95">View MathML</a> to (5) has the property mentioned above.

In conclusion, in order to apply Theorem 1, we will introduce a novel integral operator whose fixed points are also fixed points of (6) and vice versa, and thus T-periodic solutions to (3). Moreover, we will show that the zero eigenvalue of the corresponding operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a> is simple. This is the aim of the next section.

3 A novel equivalent integral operator

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M156">View MathML</a>, hence equation (7) reads as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157">View MathML</a>. Consider the integral equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M158">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M159">View MathML</a> is defined in the next lemma and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M160">View MathML</a> is a given point.

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M163">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M164">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165">View MathML</a>.

We can now formulate the following result.

Lemma 1Assume (H1)-(H3) and that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M166">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165">View MathML</a>satisfy the conditions:

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M168">View MathML</a>.

(H5) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M169">View MathML</a>.

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M170">View MathML</a>.

Define<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M171">View MathML</a>as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M172">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M173">View MathML</a>. Then (8) is equivalent to (6). Moreover, the zero eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a>is simple, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M175">View MathML</a>is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M176">View MathML</a>

Proof First of all observe that, under our assumptions, we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M177">View MathML</a>

Indeed, arguing by contradiction assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M178">View MathML</a> is such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M179">View MathML</a>

then, by the definition of ξ, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M180">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M181">View MathML</a>

thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M183">View MathML</a> are linearly dependent, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M183">View MathML</a> is also an eigenvector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M185">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M186">View MathML</a>

In conclusion, we should have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M187">View MathML</a>

On the other hand, as it is easy to verify our conditions imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M188">View MathML</a>

(9)

We now prove the equivalence between (6) and (8). Clearly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106">View MathML</a> then (8) is satisfied. Conversely, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M106">View MathML</a> is a solution to (8), hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M192">View MathML</a>

Integrating on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M194">View MathML</a>

or equivalently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M195">View MathML</a>

(10)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M196">View MathML</a> from (10), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M197">View MathML</a> and so from (8) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M157">View MathML</a>.

It remains to prove the second part of the lemma. For this, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M110">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M202">View MathML</a>. To simplify the notation in the sequel, we omit the subscript θ. Observe that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M203">View MathML</a>

Then the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M204">View MathML</a>

can be rewritten as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M205">View MathML</a>

(11)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M207">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M208">View MathML</a>

then the equation (11) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M209">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M210">View MathML</a> is an eigenvector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a> corresponding to the zero eigenvalue, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M212">View MathML</a>. Assume now that there exists an adjoint vector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M213">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M210">View MathML</a>, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M215">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M216">View MathML</a>

(12)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201">View MathML</a>. By assumption, there are no adjoint Floquet solutions to (5), thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M218">View MathML</a>

(13)

does not possess T-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M219">View MathML</a>. The integral form of (13) is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M220">View MathML</a>

(14)

Therefore, it remains to show that (12) and (14) coincide, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M221">View MathML</a>

(15)

For this, integrating (12) on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M223">View MathML</a>

that is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M224">View MathML</a>

On the other hand <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M196">View MathML</a>, thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M226">View MathML</a>

(16)

From (16), we get the following form for (15)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M227">View MathML</a>

(17)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M228">View MathML</a>

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M229">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M230">View MathML</a>

(18)

Finally substituting (18) into (17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M231">View MathML</a>

(19)

By our definition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M233">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M234">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M235">View MathML</a>. Therefore, (19) is satisfied and this concludes the proof. □

Remark 1 Observe that a little though convinces of the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M166">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M165">View MathML</a> satisfying the conditions (H4)-(H6) of Lemma 1.

Furthermore, recall that assumptions (H1)-(H2) ensure that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M142">View MathML</a> is condensing with constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M239">View MathML</a> (see [40]); moreover, [[38], Theorem 1.5.4] guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140">View MathML</a> is also condensing with the same constant. Finally, by [[38], Theorem 2.6.11], zero turns out to be an eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M65">View MathML</a> of finite multiplicity. The second part of the proof of Lemma 1 shows that it is simple.

4 The Malkin bifurcation function

In the previous section, Lemma 1 states that the operator P, associated to the integral equation (8) satisfies the conditions of Theorem 1. This section is devoted to the computation of the following Malkin bifurcation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242">View MathML</a> associated to (8)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M243">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M53">View MathML</a> is an eigenvector of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M246">View MathML</a>

For notational convenience, we simply denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M247">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7">View MathML</a>. In order to compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242">View MathML</a>, it is necessary to determine <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7">View MathML</a> in explicit form. The following result solves the problem.

Lemma 2Assume (H1)-(H6), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M251">View MathML</a>

whereγis an eigenvector of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M252">View MathML</a>corresponding to the eigenvalue 1 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M253">View MathML</a>is the characteristic function of the interval<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M193">View MathML</a>.

Proof By assumption, E has the Radon-Nikodym property, then the eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245">View MathML</a> can be determined, without loss of generality in the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M256">View MathML</a>. Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M257">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M258">View MathML</a>

First, by using assumption (H1), we calculate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M259">View MathML</a>

Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M260">View MathML</a>

(20)

We now calculate the adjoint operator for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M261">View MathML</a>. For this, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M262">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M263">View MathML</a>

Finally, we calculate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M264">View MathML</a>, for this consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M265">View MathML</a>

i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M266">View MathML</a>

Now, we are in the position to determine the eigenvector <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M7">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M245">View MathML</a>. We have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M269">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M270">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M271">View MathML</a>, then the previous equation takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M272">View MathML</a>

(21)

namely γ is an eigenvector of the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M273">View MathML</a> corresponding to the simple eigenvalue 1. Therefore, by replacing y with γ in (20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M274">View MathML</a>

The aim now is to find γ, for this consider the adjoint equation to (5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M275">View MathML</a>

(22)

and the solution of (22), defined for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M201">View MathML</a>, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M277">View MathML</a>

(23)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M278">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M279">View MathML</a>

Since v is T-periodic, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M280">View MathML</a>

(24)

By using (24) into (23), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M281">View MathML</a>

(25)

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M282">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M283">View MathML</a> and (25) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M284">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M285">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M286">View MathML</a> is a solution to (21). Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M287">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M99">View MathML</a> is the T-periodic solution to the adjoint equation (22). Finally, from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M289">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M290">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M291">View MathML</a>

By (9), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M292">View MathML</a>

In conclusion,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M293">View MathML</a>

 □

Lemmas 1 and 2, together with the fact that (H1)-(H2) ensure the condensivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M140">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, of constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M296">View MathML</a> (see [40]) allow to apply Theorem 1 to state the following.

Theorem 2Assume (H1)-(H6). If there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M297">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M298">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M299">View MathML</a>. Then there exists a branch ofT-periodic solutions to (3) of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M300">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M4">View MathML</a>sufficiently small and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M302">View MathML</a>.

Remark 2 The function w can be calculated in an explicit form as shown in [[35], Theorem 2.3] and [[36], Lemma 3].

5 An invariance property of the Malkin bifurcation function

In what follows, we state an interesting property of the Malkin bifurcation functions introduced before. Precisely, we can prove the following result.

Proposition 1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M50">View MathML</a>, assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M304">View MathML</a>is simple. Then the Malkin bifurcation function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M305">View MathML</a>associated to system (6) coincide with the Malkin bifurcation function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M242">View MathML</a>associated to system (8).

Proof Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M307">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M308">View MathML</a>, then by (9) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M309">View MathML</a>

 □

Remark 3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M305">View MathML</a> can be rewritten in the classical form of the Malkin bifurcation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M311">View MathML</a> for ordinary differential equations as (1) of the Introduction. In fact, consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M312">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M314">View MathML</a>, Ψ is the superposition operator generated by ψ and v solves (22) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M315">View MathML</a>

6 An example

In order to introduce an example consistent with the general approach of the paper that requires the employ of the theory of condensing operators, we are led to consider partial differential equations of hyperbolic type, whose abstract formulation in Banach spaces gives rise to infinitesimal generators of noncompact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316">View MathML</a>-semigroups; see, e.g., [42].

Precisely, following [43] and [44], we present a concrete, not academic example concerning the existence of periodic solutions of a system of two autonomous damped wave equations in a thin domain with Neumann boundary conditions. The study of the dynamics of partial differential equations in thin domains has received many attention in the past few years; see [45] and the extensive references therein. The system has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M317">View MathML</a>

(26)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M319">View MathML</a>, Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M320">View MathML</a>-smooth bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M321">View MathML</a>, ν denotes the outward unit normal vector to Q, λ is a small positive parameter representing the thickness of the domain of the variable λy, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M322">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M323">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M325">View MathML</a> are positive constants and the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327">View MathML</a> are of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M328">View MathML</a> jointly in their arguments. The linear part of system (26) generates an exponentially stable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316">View MathML</a>-semigroup in a suitable Banach space; see [43] as well as the related references therein. Under the assumption of the existence of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M330">View MathML</a>-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M331">View MathML</a> of the limit problem, obtained as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M332">View MathML</a>, and suitable conditions on the growth of the derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327">View MathML</a> with respect to their arguments, it is shown in [44] that [[43], Theorem 1] applies. This result guarantees the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M335">View MathML</a> such that, for fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M336">View MathML</a>, system (26) has an isolated <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M337">View MathML</a>-periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M338">View MathML</a>. The crucial assumption of [[43], Theorem 1] is that the zero eigenvalue of the linearized system around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M339">View MathML</a> is simple. For a single damped wave equation of system (26) with the nonlinear term ϕ depending periodically on time t, the existence of periodic solutions was studied in [46].

Consider now a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M337">View MathML</a>-periodic perturbation of (26) of small amplitude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M341">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M342">View MathML</a>

(27)

If we assume that the superposition operators generated by the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M326">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M327">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M345">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M346">View MathML</a> satisfy assumption (H2) of this paper, then (H1) and [[47], Theorem 4.3.1] ensure that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M316">View MathML</a>-semigroup generated by the linearization around <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M348">View MathML</a> of the unperturbed system, corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M40">View MathML</a> in (27), is strongly contractive with respect to the Hausdorff measure of noncompactness <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/101/mathml/M84">View MathML</a>, i.e., χ-strongly contractive. Therefore, our abstract bifurcation result Theorem 2 applies to system (27).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors wrote this paper in collaboration and with the same responsibility. All authors read and approved the final version of the manuscript.

Acknowledgements

Dedicated to professor Jean Mawhin on the occasion of his seventieth birthday.

The first two authors acknowledge the support by RFBR Grants 10-01-93112-a and 12-01-0392-a. The third one acknowledges the support by the GNAMPA of the Istituto di Alta Matematica. The authors would like also to thank the referees for their helpful comments and suggestions which improved the presentation of the paper.

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