Existence and Nonexistence of Solutions for Fourth-Order Nonlinear Difference Boundary Value Problems via Variational Methods

1College of Continuing Education and Open College, Guangdong University of Foreign Studies, Guangzhou 510420, China 2Science College, Hunan Agricultural University, Changsha 410128, China 3School of Economics and Management, South China Normal University, Guangzhou 510006, China 4Modern Business and Management Department, Guangdong Construction Polytechnic, Guangzhou 510440, China 5School of Mathematics and Statistics, Central South University, Changsha 410083, China


Introduction
Throughout this paper, we denote by N, Z, R the sets of all natural numbers, integers, and real numbers, respectively.Let the symbol * denote the transpose of a vector.For any integers  and  with  ≤ , [, ] Z is defined by the discrete interval {,  + 1, . . ., }.Now, we are concerned with the existence and nonexistence solutions to the fourth-order nonlinear difference equation As usual, a solution of (1), (2), in other words, a function  : [−1,  + 2] Z → R, satisfies both (1) and (2).
We may think of boundary value problem (BVP) (1), (2) as being a discrete analogue of the following fourth-order nonlinear differential equation:
Domshlak and Matakaev [17] in 2001 investigated the oscillation properties of the delay difference equation for  = 2 and  = 3 near the 2-periodic critical states with respect to its oscillation properties.By making use of "the Sturmian comparison method: discrete version", they obtained some conditions for the existence and for the nonexistence of eventually positive solution.
In 2010, He, Yang and Yang [18] considered the following second-order three-point discrete boundary value problem: By using the topological degree theory and the fixed point index theory, they provided sufficient conditions for the existence of sign-changing solutions, positive solutions, and negative solutions.
Investigating the high order difference equation Leng [22] established some new criteria for the existence and multiplicity of periodic and subharmonic solutions of (10) based on the linking theorem in combination with variational technique.
In this paper, we shall study the boundary value problem for a fourth-order nonlinear difference equation ( 1), (2).Via variational methods and critical point theory, sufficient conditions are obtained for the existence of at least two nontrivial solutions, the existence of  distinct pairs of nontrivial solutions, and nonexistence of solutions.The motivation for the present work stems from the recent papers [3,5].
Throughout the whole paper, we suppose that there exists a function (, ) such that for any and The remainder of this article is organized as follows.In Section 2, we shall give some preliminary lemmas and establish the variational structure of BVP (1), (2).In Section 3, we shall give sufficient conditions to the existence and nonexistence solutions.In Section 4, we shall complete the proofs of the main results.Some examples illustrating our main results are given in Section 5.
For the basic knowledge of variational methods, the reader is referred to [30][31][32].

Preliminary Lemmas
Assume that  is a real Banach space and  ∈  1 (, R) is a continuously Fréchet differentiable functional defined on .As usual,  is said to satisfy the Palais-Smale condition if any sequence {  } ∞ =1 ⊂  for which {(  )} ∞ =1 is bounded and   (  ) → 0 as  → ∞ possesses a convergent subsequence.Here, the sequence {  } ∞ =1 is called a Palais-Smale sequence.Let  be a real Banach space.We denote by the symbol   the open ball in  about 0 of radius ,   its boundary, and   its closure.
In the present article, we define a vector space  by and for any  ∈ , define and Remark 1.For any  ∈ , it is easy to see that As the case stands,  is isomorphic to R  .In the following and in the sequel, when we write  = ((1), (2), . . ., ()) ∈ R  , we always imply that  can be extended to a vector in  so that ( 19) is satisfied.
For any  ∈ , let the functional  be denoted by Then  ∈  1 (, R) and Thus,   () = 0 if and only if Thereupon a function  ∈  is a critical point of the functional  on  if and only if  is a solution of BVP ( 1), (2).

Proofs of the Main Results
In this section, we shall finish proofs of the main results via variational methods.
From the proof of (36), we have that () is bounded from above in .Denote As a consequence, on the one hand, there is a sequence {  } in  such that By (27), on the other hand, the functional () satisfies (40) means that lim ‖‖→+∞ () = −∞ which implies that {  } is bounded.As a result, {  } has a convergent subsequence in  denoted by {   }.Let By the reason of the continuity of () in , it is easy to see that ( 0 ) =  0 .That is,  0 ∈  is a critical point of ().
Remark 10.From the course of the proof of Theorem 4, the conclusion of Corollary 5 is evidently correct.
Remark 11.The techniques of the proof of Theorem 6 are just the same as those carried out in the proof of Theorem 4. We do not repeat them here.
Remark 12.According to Theorem 6, it is easy to see that the conclusion of Corollary 7 is true.
Proof of Theorem 8 .For any  ∈ [1, ] Z , by the continuity of (, ) in , () can be viewed as a continuously differentiable functional defined on .It comes from () and ( 1 ) that (0) = 0. Owing to the condition (), () is even.From the process of proof of Theorem 4, () is bounded from below and satisfies the Palais-Smale condition.Next, in the light of Clark Theorem, we shall find a set Ω and an odd map such that Ω is homeomorphic to  −1 by an odd map. Set It is obvious that Ω is homeomorphic to  −1 by an odd map.(44) implies that sup Ω (−()) < 0. As a result of Clark Theorem, () has at least  distinct pairs of nonzero critical points.As a consequence, BVP (1), (2) has at least  distinct pairs of nontrivial solutions.The desired result is obtained.
Proof of Theorem 9 .On the contrary, we suppose that BVP (1), ( 2) has a nontrivial solution.Therefore, () has a nonzero critical point x.
On the other hand, by (), we have which is a contradiction with (52).

Some Examples
In this section, we shall provide three examples to illustrate our main results.
satisfying the boundary value conditions We have It is easy to verify that all the suppositions of Theorem 9 are satisfied and then BVP (66), (67) has no nontrivial solutions.