Periodic Solutions for Semilinear Fourth-Order Differential Inclusions via Nonsmooth Critical Point Theory

Three periodic solutions with prescribed wavelength for a class of semilinear fourth-order differential inclusions are obtained by using a nonsmooth version critical point theorem. Some results of previous related literature are extended. 1. Introduction The Fisher-Kolmogorov (FK for short) equation was proposed as a model for phase transitions and other bistable phenomena and is one of the most fundamental models in mathematical biology and ecology; for example, see Zimmerman [1], Coullet et al. [2], and Dee and SaarLoos [3]. Recently, many authors were interested in extended FK equation of the form where is a positive constant and and are continuous positive -periodic functions on . Equation (1) arises as the mesoscopic model of a phase transition in a binary system near the Lipschitz point [4, 5], and (1) is frequently used as a model for the study of pattern formation in an unstable spatially homogeneous state [2, 3]. It has been attracting more and more attention due to its significant value in theory and practical application [6–9]. Especially, in [10], by using the symmetric mountain-pass theorem, Ma and Dai considered the nonlocal semilinear fourth-order differential equation They obtained the existence of infinitely many distinct pairs of solutions of the above problem, where is a positive constant, is a positive continuous even and -periodic function, and is continuous and monotone decreasing. It is worth mentioning that the method used in [10] is not valid for more general nonlinearity. Furthermore, to the best of our knowledge, there is no author using the nonsmooth version critical point theory to consider the extended FK equation. In view of this, this paper is concerned with the existence of three periodic solutions of the following semilinear fourth-order differential inclusion: where(H1) are real parameters, is a positive constant, and is a positive continuous even -periodic function on ; ？ is a locally Lipschitz function defined on satisfying(F1) , for ,(F2)there exist constants and , such that (F3) where and are two given positive constants. is defined on satisfying(G1) is measurable for each , is locally Lipschitz for , and and for a.e. and ;(G2)there exists constant , such that where is defined in (F2). By applying a nonsmooth version critical point theorem [11, 12], we prove that, when and are in given interval, (3) admits at least three solutions. Moreover, we achieve an estimate of the solutions norms independent of , and . Concretely, we get the following main result. Theorem 1. Assume that (H1) and (F1)–(F3) hold. Then there