Existence of periodic solutions of fourth-order nonlinear difference equations

By making use of the critical point theory, the existence of periodic solutions for fourth-order nonlinear difference equations is obtained. The proof is based on the Saddle Point Theorem in combination with variational technique. The problem is to solve the existence of periodic solutions of fourth-order nonlinear difference equations. Results obtained complement the existing one.


Introduction
Let N, Z and R denote the sets of all natural numbers, integers and real numbers respectively. For any a, b ∈ Z, define Z(a) = {a, a + 1, . . .}, Z (a, b) = {a, a + 1, . . . , b} when a ≤ b. Let the symbol * denote the transpose of a vector.
The present paper considers the following fourth-order nonlinear difference equation 2 r n−2 2 u n−2 = f (n, u n+1 , u n , u n−1 ), n ∈ Z, (1.1) where is the forward difference operator u n = u n+1 − u n , 2 u n = ( u n ), r n > 0 is real valued for each n ∈ Z, f ∈ C(Z × R 3 , R), r n and f (n, v 1 , v 2 , v 3 ) are T -periodic in n for a given positive integer T . We may think of (1.1) as a discrete analogue of the following fourth-order functional differential equation (1.2)

Equation (1.2) includes the following equation
u (4) (t) = f (t, u(t)), t ∈ R, (1.3) which is used to model deformations of elastic beams [9,29]. Equations similar in structure to (1.2) arise in the study of the existence of solitary waves of lattice differential equations, see Smets and Willem [31].
In 1995, Peterson and Ridenhour [27] considered the disconjugacy of the following equation 4 u n−2 + q n u n = 0, n ∈ Z.
Yan, Liu [34] in 1997 and Thandapani, Arockiasamy [32] in 2001 studied the following fourth-order difference equation of form, 2 r n 2 u n + f (n, u n ) = 0, n ∈ Z. (1.4) the authors obtain criteria for the oscillation and nonoscillation of solutions for Eq. (1.4). When β > 2, in Theorem 1.1, Cai et al. [7] have obtained some criteria for the existence of periodic solutions of the following fourth-order difference Eq.
Furthermore, [7] is the only paper we found which deals with the problem of periodic solutions to fourth-order difference Eq. (1.5). When β < 2, can we still find the periodic solutions of (1.5)? By using various methods and techniques, such as Schauder fixed point theorem, the cone theoretic fixed point theorem, the method of upper and lower solutions, coincidence degree theory, a series of existence results of nontrivial solutions for differential equations have been obtained in [4,6,8,20]. Critical point theory is also an important tool to deal with problems on differential equations [9,11,12,24,29,33]. Because of applications in many areas for difference equations [1,21,25], recently, a few authors have gradually paid attention to applying critical point theory to deal with periodic solutions on discrete systems, see [16][17][18]30,36,38]. Particularly, Guo and Yu [16][17][18] and Shi et al. [30] studied the existence of periodic solutions of second-order nonlinear difference equations by using the critical point theory. Compared to one-order or second-order difference equations, the study of higher-order equations, and in particular, fourth-order equations, has received considerably less attention(see, for example, [1,7,10,14,21,27,28,32,34] and the references contained therein). However, to the best of our knowledge, results obtained in the literature on the periodic solutions of (1.1) are very scarce. Since f in (1.1) depends on u n+1 and u n−1 , the traditional ways of establishing the functional in [16][17][18]36,38] are inapplicable to our case. The main purpose of this paper is to give some sufficient conditions for the existence of periodic solutions to fourth-order nonlinear difference equations. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. In particular, our results complement the result in the literature [7]. In fact, one can see the following Remark 1.4 for details.
For basic knowledge on variational methods, we refer the reader to [15,24,26,29]. Let Now we state the main results of this paper.
Theorem 1.1 Assume that the following hypotheses are satisfied: Then for any given positive integer m > 0, (1.1) has at least one mT -periodic solution.
Remark 1.2 Assumption (F 4 ) implies that for each n ∈ Z there exist constants a 3 > 0 and

Remark 1.3
The results of Theorems 1.1 and 1.2 ensure that (1.1) has at least one mTperiodic solution. However, in some cases, we are interested in the existence of nontrivial periodic solutions for (1.1).
In this case, we have Then for any given positive integer m > 0, (1.1) has at least one nontrivial mT -periodic solution.
Then for any given positive integer m > 0, (1.1) has at least one nontrivial mT -periodic solution.
Theorem 1.5 Assume that the following hypotheses are satisfied: (F 13 ) there exist constants a 7 > 0 and γ , 1 < γ ≤ α such that Then for any given positive integer m > 0, (1.5) has at least one nontrivial mT -periodic solution.
Then for any given positive integer m > 0, (1.5) has at least one nontrivial mT -periodic solution.
Remark 1.4 When β > 2, in Theorem 1.1, Cai et al. [7] have obtained some criteria for the existence of periodic solutions of (1.5). When β < 2, we can still find the periodic solutions of (1.5). Hence, Theorems 1.5 and 1.6 complement the existing one. The rest of the paper is organized as follows. First, in Sect. 2, we shall establish the variational framework associated with (1.1) and transfer the problem of the existence of periodic solutions of (1.1) into that of the existence of critical points of the corresponding functional. Some related fundamental results will also be recalled. Then, in Sect. 3, we shall complete the proof of the results by using the critical point method. Finally, in Sect. 4, we shall give two examples to illustrate the main results.

Variational structure and some lemmas
In order to apply the critical point theory, we shall establish the corresponding variational framework for (1.1) and give some lemmas which will be of fundamental importance in proving our main results. We start by some basic notations.
Let S be the set of sequences u = (. . . , u −n , . . . , For any u, v ∈ S, a, b ∈ R, au + bv is defined by Then S is a vector space. For any given positive integers m and T , E mT is defined as a subspace of S by Clearly, E mT is isomorphic to R mT . E mT can be equipped with the inner product by which the norm · can be induced by It is obvious that E mT with the inner product (2.1) is a finite dimensional Hilbert space and linearly homeomorphic to R mT .
On the other hand, we define the norm · s on E mT as follows: for all u ∈ E mT and s > 1.
Since u s and u 2 are equivalent, there exist constants c 1 , c 2 such that c 2 ≥ c 1 > 0, and Clearly, u = u 2 . For all u ∈ E mT , define the functional J on E mT as follows: where Clearly, J ∈ C 1 (E mT , R) and for any u = {u n } n∈Z ∈ E mT , by using u 0 = u mT , u 1 = u mT +1 , we can compute the partial derivative as ∂ J ∂u n = − 2 r n−2 2 u n−2 + f (n, u n+1 , u n , u n−1 ).
Thus, u is a critical point of J on E mT if and only if 2 r n−2 2 u n−2 = f (n, u n+1 , u n , u n−1 ), ∀n ∈ Z(1, mT ).
Let V be the direct orthogonal complement of E mT to W , i.e., E mT = V ⊕ W . For convenience, we identify u ∈ E mT with u = (u 1 , u 2 , . . . , u mT Let E be a real Banach space, J ∈ C 1 (E, R), i.e., J is a continuously Fréchetdifferentiable functional defined on E. J is said to satisfy the Palais-Smale condition (P.S. condition for short) if any sequence u (k) ⊂ E for which J u (k) is bounded and J u (k) → 0(k → ∞) possesses a convergent subsequence in E.
Let B ρ denote the open ball in E about 0 of radius ρ and let ∂ B ρ denote its boundary.
and id denotes the identity operator. Let On the other hand, we know that Since (2.8) Thus, we have The above inequality implies that v (k) is bounded. Next, we shall prove that w (k) is bounded. Since where θ ∈ (0, 1). It is not difficult to see that Since there exist z (k) ∈ R, k ∈ N, such that w (k) = z (k) , z (k) , . . . , z (k) * ∈ E mT , then  For k large enough, we have Take By (F 4 ), we have ∂v 2 v 2 with respect to the second and third variables implies that there exists a constant M 4 > 0 such that for n ∈ Z(1, mT ) and v 2 1 + v 2 2 ≤ R 1 . Therefore, By (F 5 ), we get where M 5 = 1 − α 2 a 2 mT + 1 2 mT M 4 . Combining with (2.4), we have Thus, This implies that u (k) 2 is bounded on the finite dimensional space E mT . As a consequence, it has a convergent subsequence.

Proof of the main results
In this Section, we shall prove our main results by using the critical point method.
Proof of Theorem 1.1 By Lemma 2.2, we know that J satisfies the P.S. condition. In order to prove Theorem 1.1 by using the Saddle Theorem, we shall prove the conditions (J 1 ) and (J 2 ).
From (2.8) and (F 2 ), for any v ∈ V , Therefore, it is easy to see that the condition (J 1 ) is satisfied.
In the following, we shall verify the condition (J 2 ). For any w ∈ W , w = (w 1 , w 2 , . . . , w mT ) * , there exists z ∈ R such that w n = z, for all n ∈ Z(1, mT ). By (F 3 ), we know that there exists a constant R 0 > 0 such that F(n, z, z) > 0 for n ∈ Z and |z| > R 0 So we have The conditions of (J 1 ) and (J 2 ) are satisfied. By (F 5 ), Combining with (F 4 ), (2,4) and (2.8), for any v ∈ V , we get, like before, Let μ = −a 2 mT , since 1 < α < 2, there exists a constant ρ > 0 large enough such that Thus, by Lemma 2.1, (1.1) has at least one mT -periodic solution.
By the Saddle Point Theorem, there exists a critical pointū ∈ E mT , which corresponds to a mT -periodic solution of (1.1).

Examples
As an application of the main theorems, finally, we give two examples to illustrate our results.
Example 2 For all n ∈ Z, assume that 2 r n−2 2 u n−2 = 2θ u n 1 + cos 2 nπ T u 2 n+1 + u 2 n θ −1 + 1 + cos 2 (n − 1)π T u 2 n + u 2 where r n > 0 is real valued for each n ∈ Z, T is a given positive integer, r n+T = r n , 0 < θ < 2. We have Then It is easy to verify all the assumptions of Theorem 1.4 are satisfied. Consequently, for any given positive integer m > 0, (4.2) has at least one nontrivial mT -periodic solution.
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