Nontrivial Periodic Solutions to Some Semilinear Sixth-Order Difference Equations

and Applied Analysis 3 We can compute the partial derivative as ∂I ∂u t = Δ 6 u t−3 + AΔ 4 u t−2 + BΔ 2 u t−1 + u t − u 3 t , t ∈ [1, T] . (19) Then, u = {u t } t∈Z is a critical point of I(u; T) on ET; that is, I 󸀠 (u; T) = 0 if and only if Δ 6 u t−3 + AΔ 4 u t−2 + BΔ 2 u t−1 + u t − u 3 t = 0, t ∈ [1, T] . (20) By the periodicity of u t , we have reduced the existence of periodic solutions of (1) to the existence of critical points of I(u; T) on E T . For convenience, we identity u ∈ E T with u = (u 1 , u 2 , . . . , u T ) T, so we draw a conclusion as follows. Lemma 4. Suppose that u t is a critical point of the functional I(u; T); then u t is a T-periodic solution of (1). We provide some lemmas which will be needed in proofs of our main results. Lemma 5. For any x(j) > 0, y(j) > 0, j ∈ [1, n], and n ∈ Z,


Introduction
In the present paper, we deal with the following sixth-order difference equation: with  0 =   = 0, where  ≥ 2 is an integer and [1, ] denotes the discrete interval {1, 2, . . ., }.Δ is the forward difference operator defined by Δ  =  +1 −  and Δ   − = Δ −1 (Δ − ). and  are positive constants satisfying  2 < 4.By using  2 index theory in combination with variational technique, we will prove nonexistence, existence, and multiplicity of nontrivial periodic solutions to (1) under convenient assumptions on .All our results only depend on  and  and are easy to satisfy.
Periodic solution problems for difference equations have been extensively studied (see the monographs of Lakshmikantham and Trigiante [1] and of Agarwal [2]).The classical theory of difference equations employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point methods; we remark that, usually, the applications of fixed point methods yield existence results only.Recently, although many new results have been established by applying variational methods, we recall here the works of Cai and Yu [3], Guo and Yu [4], and Deng et al. [5].
The variational approach represents an important advance as it allows proving multiplicity results as well.
In general, (1) may be regarded as a discrete analogue of the following sixth-order differential equation: 6   6 +   4   4 + As to (2), it is a model for describing the behavior of phase fronts in materials that are undergoing a transition between the liquid and solid state and be widely studied; one can refer to [6][7][8] and references therein.Difference equations, the discrete analogs of differential equations, represent the discrete counterpart of ordinary differential equations and are usually studied in connection with numerical analysis.They occur widely in numerous settings and forms, both in mathematics itself and in its applications to computing, statistics, electrical circuit analysis, biology, dynamical systems, economics, and other fields.For the general background of difference equations, one can refer to monographs [2,[9][10][11][12][13] for details.
Since 2003, critical point theory has been a powerful tool to establish sufficient conditions on the existence of periodic solutions of difference equations and many significant results have been obtained; see, for example, [4,14,15].Compared to first-order or second-order difference equations, the study of 2 Abstract and Applied Analysis higher order difference equations has received considerably less attention.For example, [16] studied in the context of discrete calculus of variational functional, and Peil and Peterson [17] studied the asymptotic behaviour of solutions of (3) with   () ≡ 0 for 1 ≤  ≤  − 1.In 2007, based on Linking Theorem, [3] gave some criteria for the existence of periodic solutions of for the case where  grows superlinear at both 0 and ∞.
Results in [3] made many assumptions on  and they are not easy to verify.The aim of this paper is to apply critical point theory to deal with the periodic solution problems of (1) when it is semilinear under concise and explicit assumptions on the period .The main results of this paper are the following three theorems.
The remaining of the paper is organized as follows.In Section 2 we establish the variational framework associated with (1) and transfer the problem on the existence of periodic solutions of (1) into the existence of critical points of the corresponding functional.We also state some fundamental lemmas for later use.Then we present the detailed proofs of main results in Section 3. Finally, we exhibit a simple example to illustrate our conclusions.

Preliminaries
In order to study the periodic solutions of (1), we will state some basic notations and lemmas, which will be used in the proofs of our main results.Let For a given integer  ≥ 2,   is defined as a subspace of  by and for , V ∈   , let Then   is a finite dimensional Hilbert space with above inner product, and the induced norm is As usual, for 1 ≤  < +∞, let and its norm is defined by Define the functional  :   → R as follows: Clearly, (; ) ∈ C 1 (  , R) and for any , V ∈   one can easily check that For any  = {  } ∈Z ∈   , by using   =  + for any  ∈ Z, We can compute the partial derivative as Then,  = {  } ∈Z is a critical point of (; ) on   ; that is,   (; ) = 0 if and only if By the periodicity of   , we have reduced the existence of periodic solutions of (1) to the existence of critical points of (; ) on   .For convenience, we identity  ∈   with  = ( 1 ,  2 , . . .,   )  , so we draw a conclusion as follows.
Lemma 4. Suppose that   is a critical point of the functional (; ); then   is a -periodic solution of (1).
We provide some lemmas which will be needed in proofs of our main results.Lemma 5.For any () > 0, () > 0,  ∈ [1, ], and  ∈ Z, where  > 1,  > 1, and Lemma 6.Let  ∈   be a critical point of (; ); for every V ∈   , there hold Proof.Let  ∈   be a critical point of (; ), according to the definition of Δ and the periodicity of   and V  ; then we have Similarly, we get Let  be a real Banach space.Σ is the subset of , which is closed and symmetric with respect to 0; that is, For any  ∈ Σ, the  2 geometric index, also called genus,  of  is defined by when there exists no such finite , set () = ∞.Finally set (0) = 0.
The following lemma is trivial.
Next, let us recall the definition of Palais-Smale condition.
Lemma 8 (see [20]).Let  be a real Banach space and let  :  → R be a C 1 functional and satisfy P.S. condition.If  is bound from below, then is a critical value of .

Proofs of Main Results
With the above preparations, we will prove our main results in this section.In order to give proofs of theorems, we need the following lemmas.
From Lemmas 5 and 6, there holds and it follows Using inequality (31), we have inequality (32) is true.
To prove inequality (38), let Similar to the proof of inequality (37), we get inequality (38).Now we will give the proof of Theorem 1.
To apply Lemmas 8 and 9 to look for nontrivial solutions for (1), next we prove that (; ) satisfies P.S. condition.Lemma 12. Let  > 0 and  2 < 4, and then the functional (; ) is bounded from below on   and satisfies P.S. condition.
(46) By (38), we have and it follows And together with (32), there holds From (37), it follows and then Therefore, by (48)-(52), we get that there exists a positive constant  such that As a consequence, { () } possesses a convergence subsequence in the finite dimensional Hilbert space   and (; ) satisfies P.S. condition.This completes the proof of Lemma 12.
From Lemmas 8 and 12, there exists  = inf ∈  (; ) is a critical value of (; ), which means that there exists a critical point of (; ) on   .Next, we devote ourselves to verifying the critical point is nontrivial. Set In Theorem 1, we have shown that () = 0 if  ≤  1 .To complete the proof of Theorem 2, we will show this does not hold for large .We prove that for  >  2 , where  2 is an appropriate number defined as (8), it holds () < 0 and the corresponding critical point is nontrivial.
We can prove the multiplicity result in Theorem 3 using Lemma 9. Denote Then for every V ∈   there exist positive constants  1 and  2 such that In particular, Since  >  2 >  2 for  = 1, 2, . . ., , we have for sufficiently small .It follows where  1 is defined in Lemma 9.According to Lemma 9, (; ) has at least  geometrically distinct critical points.Furthermore,  >  2 >  2 ; similar to the proof of Theorem 2, we draw a conclusion that all  distinct critical points we have obtained are all nontrivial.And the proof of Theorem 3 is completed.
Finally, we exhibit a simple example to illustrate our conclusions.

Solution.
Here  = √ 10 > 0,  = 3 > 0,  Remark.From the given example, one can find our results only depend on coefficients  and  which are very easy to verify and rather relaxed.