Elsevier

Applied Mathematics and Computation

Volume 274, 1 February 2016, Pages 590-603
Applied Mathematics and Computation

Pseudo S-asymptotically periodic solutions of second-order abstract Cauchy problems

https://doi.org/10.1016/j.amc.2015.11.034Get rights and content

Abstract

In this work we discuss the existence of pseudo S-asymptotically periodic mild solutions for a second order abstract Cauchy problem. In particular, we show that pseudo S-asymptotically ω-periodic cosine functions of operators are ω-periodic. The paper is completed with an application to the nonautonomous wave equation.

Introduction

The existence of periodic solutions and its various extensions to the second order abstract Cauchy problem has been studied in several works. We mention here only [1], [2], [3] and the references mentioned therein.

On the other hand, in [4] the authors introduce the concept of pseudo S-asymptotically periodic function, establish some qualitative properties of this type of functions and study the existence of pseudo S-asymptotically periodic mild solutions for a class of abstract neutral differential equations of first order.

The aim of this work is to study the existence of pseudo S-asymptotically periodic mild solutions to a second order abstract Cauchy problem. Specifically, we are concerned with the existence of pseudo S-asymptotically periodic mild solutions for initial value problems modeled in a Banach space (X, ∥·∥) by x(t)=A(t)x(t)+f(t,x(t)),t0,x(0)=x0,x(0)=x1,where x(t) ∈ X, A(t): DXX for t ≥ 0 are closed linear operators that satisfy appropriate conditions and f: [0, ∞) × XX is a continuous function.

The terminology and notations are those generally used in functional analysis. In particular, if (Z, ‖·‖Z) and (Y, ‖·‖Y) are Banach spaces, we indicate by L(Z,Y) the Banach space of bounded linear operators from Z into Y endowed with the uniform operator topology, and we abbreviate this notation to L(Z) whenever Z=Y. When T is an appropriate linear map, we denote by σp(T) the point spectrum of T. Let I be a compact interval of real numbers. Along this paper, C(I, Z) is the space of continuous functions from I into Z endowed with the norm of the uniform convergence, and Cb([0, ∞), Z) is the space of bounded continuous functions from [0, ∞) into Z endowed with the norm of the uniform convergence.

This paper has four sections. In Section 2 we revise some concepts on cosine functions and pseudo S-asymptotically periodic functions needed to establish our results. We also introduce the concept of pseudo S-asymptotic periodicity for operator-valued functions and prove some relations between pseudo S-asymptotically periodicity and periodicity for the case of cosine and sine functions, see Theorem 2.1. In Section 3 we establish some results about the existence of global mild solutions for the problem (1.1)–(1.2). In the last section, an application involving the wave equation is considered.

Section snippets

Preliminaries

Next, we review some fundamental concepts needed to establish our results.

Existence of mild solutions

In this section we study the existence of pseudo S-asymptotically ω-periodic mild solutions for (1.1)–(1.2). Next we assume that the operator functions C(·) and S(·) are ω-periodic functions and M > 0 is a constant such that ‖C(t)‖ ≤ M and ‖S(t)‖ ≤ M for all tR.

Initially we consider the linear inhomogeneous problem x(t)=Ax(t)+f(t),t0,x(0)=x0,x(0)=x1,

Lemma 3.1

Assume thatfE0(X) is a function such that u(t)=0tS(ts)f(s)dsdefines a bounded function on [0, ∞). Then the mild solution of problem (3.1)

Applications

In this section, we study the problem of vibrations of a string. To simplify the exposition, we shall only consider a string located on the interval [0, π].

We consider a system described by the equations 2t2w(t,ξ)=2ξ2w(t,ξ)+b(t,ξ)ξw(t,ξ)+q(t,ξ,w(t,ξ)),t0,0ξπ,w(t,0)=w(t,π)=0,t[0,),w(0,ξ)=z0(ξ),tw(0,ξ)=z1(ξ)ξ[0,π],where b:[0,)×[0,π]R, and z0,z1:[0,π]R are functions that satisfy appropriate conditions which will be specified later, and q:[0,)×[0,π]×RR is a continuous function

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    This author was partially supported by CONICYT under grant FONDECYT 1130144 and DICYT-USACH.

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