Boundary Value Problems for Fourth Order Nonlinear p-Laplacian Difference Equations

φ p (s) = |s| p−2 s (p > 1), f ∈ C(Z(1, k) ×R,R). In the last decade, by using various techniques such as critical point theory, fix point theory, topological degree theory, and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations (see [1–7] and references therein). Among these approaches, the critical point theory seems to be a powerful tool to deal with this problem (see [5, 7–9]). However, compared to the boundary value problems of lower order difference equations ([6, 8, 10–13]), the study of boundary value problems of higher order difference equations is relatively rare (see [9, 14, 15]), especially the works by using the critical point theory [16]. For the background on difference equations, we refer to [17]. In this paper, we will consider the existence of solutions of the boundary value problem of (1) with (2). First, we will construct a functional J such that solutions of the boundary value problem (1) with (2) correspond to critical points of J. Then, by using Mountain pass lemma, we obtain the existence of critical points of J. We mention that (1) is a kind of difference equation containing both advance and retardation. This kind of difference equation has many applications both in theory and practice. For example, in [17], Agarwal considered the following difference equation:

In the last decade, by using various techniques such as critical point theory, fix point theory, topological degree theory, and coincidence degree theory, a great deal of works have been done on the existence of solutions to boundary value problems of difference equations (see [1][2][3][4][5][6][7] and references therein).Among these approaches, the critical point theory seems to be a powerful tool to deal with this problem (see [5,[7][8][9]).However, compared to the boundary value problems of lower order difference equations ([6, 8, 10-13]), the study of boundary value problems of higher order difference equations is relatively rare (see [9,14,15]), especially the works by using the critical point theory [16].For the background on difference equations, we refer to [17].
In this paper, we will consider the existence of solutions of the boundary value problem of (1) with (2).First, we will construct a functional  such that solutions of the boundary value problem (1) with (2) correspond to critical points of .Then, by using Mountain pass lemma, we obtain the existence of critical points of .We mention that (1) is a kind of difference equation containing both advance and retardation.This kind of difference equation has many applications both in theory and practice.For example, in [17], Agarwal considered the following difference equation: with the boundary value conditions as an example.It represents the amplitude of the motion of every particle in the string.And in [7], the authors considered the following second order functional difference equation: Journal of Applied Mathematics with different boundary value conditions where the operator  is the Jacobi operator given by In [18], the authors considered the second order -Laplacian difference equation: with boundary value conditions As for the periodic and subharmonic solutions of -Laplacian difference equations containing both advance and retardation, we refer to [19].And for the periodic solutions of -Laplacian difference equations, we refer to [20].Throughout this paper, we assume that there exists a function (, , V) which is differentiable in (, V) and (, 0, 0) = 0 for each  ∈ Z(0, ), satisfying for  ∈ Z(1, ).

Preliminaries and Main Results
Lemma which implies that and  * () = 1 is obvious.If  > 2, then we have which implies that and  * () = 1 is obvious.Now the proof is complete.
Before we apply the critical point theory, we will establish the corresponding variational framework for (1) with (2). Let Then  is a -dimensional Hilbert space.Obviously,  is isomorphic to R  .In fact, we can find a map  :  → R  defined by Define the inner product on  as The corresponding norm ‖ ⋅ ‖ can be induced by For all  ∈ , define the functional () on  as follows: Clearly,  ∈  1 (, R).We can compute the partial derivative as Let   denote the open ball in  with radius  and center 0, and let   denote its boundary.
In order to obtain the existence of critical points of  on , we need to use the following basic lemma, which is important in the proof of our main results.
Then  possesses a critical value  ≥  given by where Let Now we state our main results.
In view of (37) and (38), it is easy to obtain the following corollary.
For the case when  = 2, we have the following corollary for the boundary value problems of the fourth order nonlinear difference equations.Corollary 8. Assume that (, , V) satisfies the following conditions.

Proof of Theorem 5
In order to prove Theorem 5, we first establish the following lemma.
Then by (49), which means that  satisfies the condition ( 1 ) of the Mountain pass lemma.By our assumptions, it is clear that (0) = 0.In order to use Mountain pass lemma, it suffices to verify that condition ( 2 ) holds.In fact, similar to the proof of (45), we have for any  ∈ .Since  0 < , it is easy to see that there exists an  ∈  with ‖‖ >  such that (±) < 0. Thus ( 2 ) holds.
In the last part of this paper, we give an example to illustrate our results.