Periodic Solutions of Second‐Order Differential Inclusions Systems with p(t)‐Laplacian

Liang Zhang,Peng Zhang

doi:10.1155/2012/475956

Abstract

The existence of periodic solutions for nonautonomous second‐order differential inclusion systems with p(t)‐Laplacian is considered. We get some existence results of periodic solutions for system, a.e. t ∈ [0, T], , by using nonsmooth critical point theory. Our results generalize and improve some theorems in the literature.

Highlights

We suppose that F : 0, T × RN → R satisfies the following assumption: A’ F t, x is measurable in t for every x ∈ RN and locally Lipschitz in x for a.e. t ∈ 0, T, F t, 0 ∈ L1 0, T and there exist positive constants C, C0, and α ∈ 0, ∞ such that ζ ∈ ∂F t, x ⇒ |ζ| ≤ C|x|α C0, 1.2 for a.e. t ∈ 0, T and all x ∈ RN
We recall some known results in nonsmooth critical point theory, and the properties of space WT1,p t are listed for the convenience of readers
We can apply Clarke’s abstract framework to f2 with the following cast of characters: i T, T, μ : 0, T with Lebesgue measure, and let Y : RN ×RN, which is a separable Banach space with the norm | · | | · |, ii let Z : Wp t ∩ W∞, which is a closed subspace of W∞, and W∞ denotes the space of measure essentially bounded functions mapping T to Y, equipped with the usual supremum norm by Definition 2.18, iii define a functional f2 on Z by 2.59, iv the mapping t → L02 t, u0 t, u 0 t ; v1, v2 is measurable for each v1, v2 in RN × RN

Summary

Introduction

Consider the second-order system with p t -Laplacian d |ut |p t −2ut ∈ ∂F t, u t a.e. t ∈ 0, T , dt. He was able to construct a substitute for the pseudogradient vector field of the smooth theory and use it to obtain nonsmooth versions of the Mountain Pass Theorem of Ambrosetti and Rabinowitz see 9 and of the Saddle Point Theorem of Rabinowitz see 10 Chang used his theory to study semilinear elliptic boundary value problems with a discontinuous nonlinearity. There are some papers discussing existence and multiplicity of periodic solutions and subharmonic solutions for problem 1.3 and 1.4 when the potential F is just locally Lipschitz in the second variable x not continuously differentiable. The main purpose of this paper is to establish the corresponding variational structure for system 1.1 , and we get some existence results of periodic solutions for system 1.1 by using nonsmooth critical point theory. We denote by p : max0≤t≤T p t > 1 throughout this paper, and we use ·, · and | · | to denote the usual inner product and norm in RN, respectively

Basic Definitions and Preliminary Results

Main Results and Proofs of Theorems

Example