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

where p is a positive constant and a and b are continuous positive 2L-periodic functions onR. 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 attractingmore and more attention due to its significant value in theory and practical application [6–9]. Especially, in [10], by using the symmetricmountain-pass theorem, Ma and Dai considered the nonlocal semilinear fourth-order differential equation


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].
(2) 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 2-periodic function, and (, ) ≡ (| − |) : [0, ] → R + 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 2 Journal of Function Spaces with the existence of three periodic solutions of the following semilinear fourth-order differential inclusion: where (H1) ,  > 0 are real parameters,  is a positive constant, and  is a positive continuous even 2-periodic function on R;  is a locally Lipschitz function defined on R satisfying where  and  2 are two given positive constants.
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 exist a nondegenerate interval [, ] ⊂ (0, ∞) and  > 0 with the following property: for any  ∈ [, ] and any function  satisfying (G1) and (G2), there exists  > 0, such that, for  ∈ (0, ), the problem (3) admits at least three solutions with norm in  less than .
Remark 2. Note that the coefficient  is even and 2-periodic, if  is a solution of (3) and () is its antisymmetric extension with respect to  = 0: Combining with (F1) and (G1), we can see that 2-periodic extension of  over R is a 2-periodic solution of the equation The rest of this paper is arranged as follows.Section 2 contains some preliminaries of nonsmooth analysis and abstract results which are needed later.Section 3 concerns a variational method for problem (3); in the final section, we give the proof of the main result.

Preliminaries
We collect some basic notions and results of nonsmooth analysis, namely, the calculus for locally Lipschitz functionals developed by Clarke [13] and Motreanu and Panagiotopoulos [14].
Let (, ‖ ⋅ ‖  ) be a Banach space, ( * , ‖ ⋅ ‖  * ) be its topological dual, and  :  → R a functional.We recall that  is locally Lipschitz (l.L.), if, for each  ∈ , there exist a neighborhood  of  and a real number  > 0 such that If  is l.L. and  ∈ , the generalized directional derivative of  at  along the direction V ∈  is The generalized gradient of  at  is the set Then for  ∈ , () ∈ 2  * is a nonempty, convex, and  *compact subset [13, Proposition 1].We call that  has compact gradient if  maps bounded subsets of  into relatively compact subsets of  * .We say that  ∈  is a critical point of l.L. functional  if 0 ∈ ().
Lemma 8 (a particular case of [12,Theorem 14]).Let (, ‖ ⋅ ‖) be a reflexive Banach space,  ⊂ R an interval, N ∈  1 (, R) a sequentially weakly lower semicontinuous functional whose derivative is of type () + , and F :  → R an l.L. functional with compact gradient, and let  ∈ R. Assume that sup Then, there exist ,  ∈ ( < ) and  > 0 with the following property: for  ∈ [, ] and any l.L. functional G :  → R with compact gradient, there exists  > 0 such that, for every  ∈ (0, ), the functional  , = N − F − G admits at least three critical points in , with norms less than .
The main hypothesis of Lemma 8 is the minimax inequality ( 16).An easy way to have it satisfied is illustrated by the following result obtained by Ricceri [16].

Lemma 12.
If  ∈  is a weak solution of (3), then  is a solution of (3).

Proof of the Main Result
In order to prove the main result, we shall show some related lemmas.Firstly, define functionals Next, we will give some properties of those functionals.
Lemma 15.Suppose (F1) and (F2) are fulfilled.Then F :  → 2 We claim that  *  →  * .Suppose the contrary; we assume there exists  > 0 such that ‖ *  −  * ‖  * >  (choose a subsequence if necessary).Hence for every  ∈ N, we can find V  ∈  with ‖V  ‖ < 1 such that Passing to a subsequence if necessary, we can assume that Combining with (F2), we have which contradicts (40).
Proof of Theorem 1.The equivalence of ‖| ⋅ |‖  defined in (21) with the usual norm in  implies that the functional N defined in (28) is sequentially weak lower semicontinuous.By Lemma 13; N  is of type () + .By Lemmas 14 and 15, F :  → R is an l.L. functional with compact gradient.First, we verify the condition (15) in Lemma 8.By (F1), (F2), and Lebourg's mean value theorem, we have hence where  1 and  2 are positive constants, so it follows from  < 2 that and therefore, (19) holds.Then we get (16) for some  ∈ R and  ⊂ (0, ∞).