Existence of Solutions to Boundary Value Problems for a Fourth-Order Difference Equation

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

Using the direct method of the calculus of variations and the mountain pass technique, Leszczyński [20] obtained the existence of at least one and at least two solutions of the difference equation: with boundary value conditions Our purpose in this article is to use the critical point theory to explore some existence criteria of solutions to the boundary value problem (1), (2) for a fourth-order nonlinear difference equation.In a special case, a necessary and sufficient condition for the existence and uniqueness of solution is also established.Results obtained here are motivated by the recent papers [18,25]. Define The rest of this article is organized as follows.In Section 2, we shall introduce some basic notations, set up the variational framework of BVP ( 1), ( 2), and give a lemma which will be useful in the proofs of our theorems.Section 3 contains our results of at least one solution.In Section 4, we shall finish proving the main results.In Section 5, we shall provide an example to illustrate our main theorem.
For any  ∈ , define the functional  as It is easy to see that  ∈  1 (, R) and For this reason,   () = 0 when and only when Thereby a function  ∈  is a critical point of the functional  on  when and only when  is a solution of the BVP (1), (2).
Let  and  be the  ×  matrices as follows.
Let  fl  + .Therefore, we rewrite () by 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 Here, the sequence {  } ∞ =1 is called a Palais-Smale sequence.Let   be the open ball in  about 0 of radius ,   be its boundary, and   be its closure.
Lemma 2 (saddle point theorem [30]).Suppose that  is a real Banach space,  =  1 ⊕  2 , where  1 ̸ = {0} and is finite dimensional.Suppose that  ∈  1 (, R) satisfies the Palais-Smale condition and ( 1 ) there are constants ,  > 0 such that Then  admits a critical value  ≥ , where and  is defined as the identity operator.

Main Theorems
Our main theorems are as follows.
If () > 0, we obtain the theorem as follows.
If () = 0, consider the following equation: with boundary value condition (2).By a similar argument to that in Section 2, we define the functional  as where f = ((1), (2), . . ., ()) * .Hence, a function  ∈  is a critical point of the functional  on  when and only when  is a solution of the BVP ( 29), (2).
It is obvious to find that the critical point of  is just the solution of the following linear equation: Let Ω = (, f).Making use of the linear algebraic theory, we can obtain the following necessary and sufficient conditions.

Proofs of the Theorems
In this section, we shall finish proofs of the theorems by using critical point theory.
Then, we prove the condition ( 2 ) of Lemma 2. For any V ∈  1 , by Hölder inequality, we have as one finds by minimization with respect to ‖V‖.In other words, Let It follows from ( 1 ) that All the assumptions of Lemma 2 are proved.According to saddle point theorem, the proof of Theorem 3 is complete.
Proof of Theorem 4. Let {  } ∈N ⊂  be such that {(  )} ∈N is bounded and   (  ) → 0 as  → ∞.Thus, for any  ∈ N, there exists a positive constant  2 such that By ( 25), (33) and Hölder inequality, we have For any  = (, , . . ., ) * ∈  2 , let and similar to the proof of Theorem 1.1 we have Condition ( 1 ) of Lemma 2 is proved.Then, we prove the condition ( 2 ) of Lemma 2. For any V ∈  1 , by Hölder inequality, we have as one finds by maximization with respect to ‖V‖.That is to say, By ( 2 ), we have Therefore, − satisfies the condition of saddle point theorem.By Lemma 2, the proof of Theorem 4 is finished.

Examples
In this section, we shall provide two examples to illustrate our main theorem.
satisfying the boundary value conditions We have  (67) From the above argument, we see that all the suppositions of Theorem 3 are satisfied; then the BVP (63), (64) has at least one solution.

Conclusions
Difference equations, the discrete analogue of differential equations, occur widely in numerous settings and forms, both in mathematics itself and in its applications in theory and practice.The boundary value problem discussed in this paper has important analogue in the continuous case of the fourth-order differential equation.Such problem is of special significance for the study of beam equations which are used to describe the bending of an elastic beam.The problem discussed in this paper can be extended to 2thorder difference boundary value problem.