Anti-periodic solutions for nonlinear evolution equations

Cheng et al. Advances in Difference Equations 2012, 2012:165 http://www.advancesindifferenceequations.com/content/2012/1/165 RESEARCH Open Access Anti-periodic solutions for nonlinear evolution equations Yi Cheng1,2 , Fuzhong Cong1* and Hongtu Hua1,2 * Correspondence: Fundamental Department, Aviation University of Air Force, Changchun, 130022, People’s Republic of China Full list of author information is available at the end of the article Abstract 1 In this paper, we use the homotopy method to establish the existence and uniqueness of anti-periodic solutions for the nonlinear anti-periodic problem  ẋ + A(t, x) + Bx = f (t) a.e. t ∈ R, x(t + T) = –x(t), where A(t, x) is a nonlinear map and B is a bounded linear operator from RN to RN . Sufficient conditions for the existence of the solution set are presented. Also, we consider the nonlinear evolution problems with a perturbation term which is multivalued. We show that, for this problem, the solution set is nonempty and weakly compact in W 1,2 (I, RN ) for the case of convex valued perturbation and prove the existence theorems of anti-periodic solutions for the nonconvex case. All illustrative examples are provided. Keywords: anti-periodic solution; evolution equation; Leray-Schauder alternative theorem; measurable selection; continuous selection 1 Introduction Anti-periodic problems have important applications in auto-control, partial differential equations and engineering, and they have been studied extensively in the past ten years. For example, anti-periodic trigonometric polynomials are important in the study of interpolation problems [], and anti-periodic wavelets are discussed in []. Recently, antiperiodic boundary conditions have been considered for the Schrödinger and Hill differential operator [, ]. Also, anti-periodic boundary conditions appear in the study of difference equations [, ]. Moreover, anti-periodic boundary conditions appear in physics in a variety of situations, see [–]. The study of anti-periodic solutions for nonlinear evolution equations was initiated by Okochi []. Since then, many authors have been devoted to investigation of the existence of anti-periodic solutions to nonlinear evolution equations in Hilbert spaces. For the details, see [–] and the references therein. In [], Chen studied the anti-periodic solution for the following first-order semilinear evolution equation: ⎧ ⎨u̇ + Au(t) + f (t, u) =  a.e. t ∈ R, ⎩u(t + T) = –u(t), (.) © 2012 Cheng 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. Cheng et al. Advances in Difference Equations 2012, 2012:165 http://www.advancesindifferenceequations.com/content/2012/1/165 Page 2 of 15 where A : RN → RN is a matrix, f : R × RN → RN is a continuous function satisfying f (t + T, u) = –f (t, –u) for all (t, u) ∈ R × RN . Here they assume that f (t, u) is a uniform bound  with respect to u and T A  < . We do not need these assumptions and consider the following semilinear anti-periodic problem: ⎧ ⎨ẋ + A(t, x) + Bx = f (t) a.e. t ∈ R, ⎩x(t + T) = –x(t), (.) where A : RN → RN is a hemicontinuous function satisfying A(t + T, x) = –A(t, –x), f : R → RN is a measurable function satisfying f (t +T) = –f (t) for all t ∈ R and B is a bounded linear operator from RN to RN . We will establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions of Eq. (.) by the theory of topological degree. In addition, we also consider the following nonlinear evolution inclusion problem: ⎧ ⎨ẋ + A(t, x) + Bx ∈ F(t, x) a.e. t ∈ I, ⎩x(T) = –x(), (.) where I = [, T]. We refer the reader to the work of [, ]. These works focused on the problem in which the multivalued term F(t, x) is an even lower semi-continuous convex function with a compact assumption. But, in this paper, we prove the existence theorems of anti-periodic solutions for the cases of a convex and of a nonconvex valued perturbation term which is multivalued based on the techniques and results of the theory of set-valued analysis and the Leray-Schauder fixed point theorem. As far as we know, there are few papers which deal with this type of anti-periodic problems. For recent developments involving the existence of anti-periodic solutions of differential equations, inequalities and other interesting results on anti-periodic boundary value problems, the reader is referred to [–] and the references therein. On the one hand, it is well known that the neural networks have been successfully applied to signal and image processing, pattern recognition and optimization. However, many neural networks with discontinuous neuron activation functions appear in the theoretical study on dynamics of neural networks, see [, ]. In order to solve some practical engineering problems, people also need to present new neural networks with discontinuous activation functions. Therefore, developing a new class of neural networks with discontinuous neuron activation functions and giving the conditions of the stability are very valuable in both theory and practice. Motivated by the above discussions, in this paper, we present a class of neural networks with discontinuous neuron activation functions. Based on our results, the existence and uniqueness of the equilibrium point is investigated. On the other hand, it has been well recognized that differential inclusions, which are certainly of their own interest, provide a useful generalization of control systems governed by differential/evolution equations with control parameters ẋ = f (t, x, u), u ∈ U(t, x), (.) where the control sets U(·, ·) may also depend on the state variable x. Let F(t, x) = f (t, x, U(t, x)). Then Eq. (.) is reduced to ẋ ∈ F(t, x), which is a particular case of the Cheng et al. Advances in Difference Equations 2012, 2012:165 http://www.advancesindifferenceequations.com/content/2012/1/165 inclusion relation in Eq. (.). Hence, we present an example of a nonlinear anti-periodic distributed parameter control system with a priori feedback for our results. This paper is organized as follows. In Section , we state some basic knowledge from multivalued analysis. In Section , we first establish the existence of anti-periodic solutions for an evolution equation by the theory of topological degree, and then, by applying the Leray-Schauder fixed point theorem, we prove the existence of anti-periodic solutions for convex and nonconvex cases. Finally, two examples for our results are presented in Section . 2 Preliminaries For convenience, we introduce some notations as follows. In Euclidean space, (·, ·) expresses an inner product, while | · | expresses the Euclidean norm. Let L ([, T]; RN ) deT note the set of the map x : [, T] → R (...truncated)