Multiple periodic solutions for discrete boundary value problem involving the mean curvature operator

: In this article, by using critical point theory, we prove the existence of multiple T - periodic solutions for di ﬀ erence equations with the mean curvature operator: where (cid:2) is the set of integers. As a T - periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well - known periodic problems. Finally, we give one example to illustrate our main results.


Introduction
Let and be the sets of integers and real numbers, respectively. For In this article, we consider the following nonlinear difference equations with mean curvature operator where λ is a positive real parameter, Δ is the forward difference operator defined by ( ) ( ) ( ) = + − u t u t u t Δ 1 , T is an integer, ( ) → + q t : and is T -periodic function, ( ) ( ) ⋅ ∈ f t C , , 1 satisfies ( ) = f t, 0 0 for each ∈ k , , , and ϕ c is the mean curvature operator defined by ( ) which has also been investigated by many authors. For example, see [1,3,4] and the references therein. If = κ 0, ( ) = q t 0 for ∈ t and = λ 1, then problem (1) is degenerated to The existence and multiplicity results of periodic and subharmonic solutions for problem (3) have been obtained by means of the variational methods in [18].
In addition, if = κ 0 and = λ 1, many authors considered the existence results for the following difference equation by means of the variational methods, For example, Yu et al. in [23] studied the existence of periodic solution of problem (4), where they considered that the nonlinear term was unbounded or bounded, respectively. Assume that N is a positive integer. If = κ 1, ( ) = q t 0 for ( ) ∈ t N 1, , Zhou and Ling [11] first used the critical point theory to study the following discrete boundary value problem: They proved the existence of infinitely many positive solutions of problem (5) under the suitable oscillating behavior of the nonlinear term f at infinity. In this article, our main aim is to use the critical point theory to establish the existence of multiple T -periodic solutions of problem (1). We consider that problem (1) is a T -periodic problem, and hence, problem (1) reduces to the following periodic boundary value problem: Obviously, problem (6) is a more general difference equation with mean curvature operator. Although many excellent results have been worked out on the existence of periodic solutions for difference equations [18][19][20][21]23], the multiple periodic solutions of the discrete boundary value problem involving the mean curvature operator show that no similar results were obtained in the literature. This article is organized as follows. In Section 2, we present some definitions and results of the critical point theory. We establish the variational framework of problem (6) and transfer the existence of periodic solutions of problem (6) into the existence of critical points of the corresponding functional. In Section 3, we obtain the existence and multiple periodic solutions for boundary value problem (6), and we give an example to illustrate our main results.

Preliminaries
In this section, our aim is to establish the existence of multiple periodic solutions of problem (6) by means of critical point theory. First, we recall some basic definitions and known results from the critical point theory.
Consider the T -dimensional Banach space: Let E be a real Banach space and We say J satisfies the Palais-Smale condition (P.S. condition) if any P.S. sequence for J possesses a convergent subsequence in E.
Let E be a finite dimensional real Banach space and → J E : λ be a function satisfying the following structure hypothesis: ( ) H Assume that λ is a real positive parameter.
The following lemma will be used to prove our main results.
Lemma 2.1. [22] Assume that ( ) H and the following conditions hold, satisfies the P.S. condition and it is bounded from below; ( ) Then, for , J λ has at least three critical points.
For every ∈ u S, put and we can compute the Frećhet derivative as follows: . It is clear that the critical points of J λ are the solutions of problem (6). . We shall prove the sequence { } u n is bounded.
If not, we assume ‖ ‖ → +∞ u n as → ∞ n . From the condition ( ) i , we take where ( ) as . This contradicts the fact | ( )| ≤ J u C λ n . Thus, the sequence { } u n is bounded in S and the Bolzano-Weierstrass theorem implies that { } u n has a convergent subsequence. In fact, Then J λ is coercive on S. The proof is complete. □ Remark 2.1. Since S be a finite dimensional real Banach space, if J λ is coercive on S, then the conclusion ( ) a 1 of Lemma 2.1 holds. From the condition ( ) i , the nonlinear term can be unbounded or bounded, and it does not require any asymptotic condition or a superlinear growth at infinity. It is different from the conditions of the literature [23].

Put
Theorem 3.1. Assume that the condition ( ) i is satisfied and there exist two positive constants c and d with , problem (6) admits at least three periodic solutions.
Proof. Our aim is to apply Lemma 2.1 to prove our conclusion. We take Φ and Ψ defined as in (7) on the space S. It is easy to verify that Φ and Ψ satisfy the assumptions required in ( ) H . In addition, it follows from Lemma 2.2 that ( ) a 1 of Lemma 2.1 hold. It remains to verify ( ) a 2 .
Put , then we have Let ( ) = w t d for every By (10), we have ( ) ( ) < φ r φ r 1 2 . The proof is complete. □ is a continuous odd function and the conditions of Theorem 3.1 hold. Then problem (6) admits at least five periodic solutions. In fact, let u be one solution of problem (6). We see that −u is also solution of problem (6) when ( ) ⋅ f t, is odd. If the conditions of Theorem 3.1 are satisfied and = u 0 is the trivial solution of problem (6), then problem (6) admits at least four different nontrivial periodic solutions and one zero solution. Moreover, if = u 0 is not the solution of problem (6), then problem (6) admits at least six different nontrivial periodic solutions. , problem (6) admits at least three periodic solutions.
Proof. In fact, in view of < < c d 0 and (13) holds, we obtain , problem (6) admits at least three periodic solutions.