Periodic and subharmonic solutions for a 2\begin{document}$n$\end{document}th-order \begin{document}$\phi_c$\end{document}-Laplacian difference equation containing both advances and retardations

We consider a 2 \begin{document}$n$\end{document} th-order nonlinear difference equation containing both many advances and retardations with \begin{document}$\phi_c$\end{document} -Laplacian. Using the critical point theory, we obtain some new and concrete criteria for the existence and multiplicity of periodic and subharmonic solutions in the more general case of the nonlinearity.

(1.2) As it is known, difference equations are widely used, not only in mathematics itself but also in its applications to other fields, such as computing, economics, biology, dynamical systems and so on.
In 2003, Guo and Yu first used the critical point theory to establish sufficient conditions for the existence of periodic solutions of difference equations. In fact, they [6,7,8] demonstrated sufficient conditions on the existence of periodic solutions of second-order nonlinear difference equations. Since then, the study of periodic solutions for difference equations has gained more attention.
In particular, Mawhin [15] considered T -periodic solutions of systems of difference equations of the form When the most of documents are referring to lower-order difference equations, the study of higher-order equations has been taken into account. In 2007, Cai and Yu [3] considered the existence of periodic solutions of the following 2nth-order nonlinear difference equation (1.4) In 2010, Zhou [27] extended f in (1.4) into super-linearity, sub-linearity and asymptotically linearity when Cai and Yu [3] considered f grows superlinearly at both zero and infinity. Moreover, a necessary and sufficient condition for the existence of the unique periodic solution of it is also established. On the other hand, the study of difference equations contain both advance and retardation which has important background and meaning in the field of cybernetics and biological mathematics has received considerably attention. For example, Liu [11] in 2014 obtained some new sufficient conditions on the existence and multiplicity of periodic solutions of the following difference equation ∆ 2 r n−2 ∆ 2 u n−2 = f (n, u n+1 , u n , u n−1 ), n ∈ Z. (1.5) Next year, by using Mountain pass lemma, Zhou and Su [25] obtained some sufficient conditions for the first time for the existence of solutions of the 2nth-order φ c -Laplacian difference equation with the boundary value conditions where n and T are given positive integer with n < T .
In the known literature, the results on periodic solutions of nonlinear difference equations containing both many advances and retardations with φ c -Laplacian are very scare. A special case of (1.1) with τ = 1 was considered in [9], like most existing literature, the authors assume that the nonlinear functional F grows quadratically at infinity. Some new criteria for the existence and multiplicity of periodic solutions was deduced in [9]. However, there are some errors in the derivation process of his main results. He used the following lemma holds, where λ(k) = 4 sin 2 π 2(k + 1) ,λ(k) = 4 sin 2 kπ 2(k + 1) , which is applied to solving boundary value problems rather than periodic solutions of difference equations. Actually, if we let u k ≡ 1 for k ∈ Z, (1.7) does not hold.
To the best of our knowledge, there is almost no result on the existence of periodic solutions of (1.1) when F grows asymptotically linearly at infinity. To fill this gap, this paper gives some new sufficient conditions for the existence and multiplicity of periodic and subharmonic solutions to (1.1) in the more general case of the nonlinear functional F . Let Our main result is as follows.
Theorem 1.1. Assume that the following hypotheses are satisfied: where λ min and λ max are given in (2.2) below. Then for any given positive integer m, (1.1) possesses at least three mT -periodic solutions.

PENG MEI, ZHAN ZHOU AND GENGHONG LIN
Note that the former (T 3 ) is much weaker than the latter (F 3 ) in the conditions that the nonlinear functional F needs to be satisfied. Thus our result improves that of Leng. In fact, if τ = 1, Theorem 1.1 reduces to a improved and correct version of Theorem 1.1 in [9]. It is worth pointing out that our sufficient conditions are based on the limit superior and limit inferior, which are more applicable. Moreover, we also extend the conclusions to a more general form. For the sake of clarity, we put the remaining results at the end of the article.
An outline of this paper reads as follows. In Section 2 we 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. In Section 3, some related fundamental results are recalled for convenience, and some lemmas are proven. Then, we complete the proof of our main result by using Linking Theorem in Section 4. Eventually, in Section 5, we illustrate our results with an example.
2. Variational structure and some lemmas. The purpose of this section is to establish the corresponding variational framework for (1.1) and cite some basic conclusions for the coming discussion.
Let S be the set of all two-side sequences, that is 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 , we define the subspace E mT of S as It is trivial to show that, E mT is isomorphic to R mT and can be endowed with the inner product Define the functional J on E mT as Clearly, J ∈ C 1 (E mT , R) and by the fact that u 0 = u mT , u 1 = u mT +1 , after a careful computation, we find ∂J ∂u k = (−1) n ∆ n (r k−n φ c (∆ n u k−n ))−f (k, u k+τ , · · · , u k , · · · , u k−τ ), ∀k ∈ Z(1, mT ).
Thus, u is a critical point of J on E mT if and only if (1.1) holds. That shows that we reduce the existence of periodic solutions of (1.1) to the existence of critical points of J on E mT . Indeed, u ∈ E mT can be identified with u = (u 1 , u 2 , · · · , u mT ) * , where * denotes the transpose of the vector. Let P be the corresponding mT ×mT matrix to the quadratic form mT k=1 (∆u k ) 2 with u k+mT = u k for k ∈ Z, which is defined by By matrix theory, we obtain that the eigenvalues of P are λ j = 2 1 − cos 2j mT π , j = 0, 1, 2, · · · , mT − 1.
3. Some results and lemmas. To make it easy for readers to scan this paper, we give some basic notations and some known results about the critical point theory.
Definition 3.1. Let E be a real Banach space, J ∈ C 1 (E, R), i.e., J is a continuously Fréchet-differentiable functional defined on E. If any sequence u (i) ⊂ E for which J u (i) is bounded and J u (i) → 0(i → ∞) possesses a convergent subsequence, then we say J satisfies the Palais-Smale condition (P.S. condition for short).
Let B ρ denote the open ball in E about 0 of radius ρ and let ∂B ρ denote its boundary.
Lemma 3.1 (Linking Theorem [16]). Let E be a real Banach space, E = E 1 ⊕ E 2 , where E 1 is finite dimensional. Suppose that J ∈ C 1 (E, R) satisfies the P.S. condition and (J 1 ) There exist constants a > 0 and ρ > 0 such that J| ∂Bρ∩E2 a; (J 2 ) There exists an e ∈ ∂B 1 ∩ E 2 and a constant R 0 ρ such that J| ∂Q 0, where Then we prove some lemmas which are useful in the proof of our main results. First, similarly to the derivation of [3], we can find the following lemma Lemma 3.2. For any u ∈ V , one has λ n min u 2 ∆ n u 2 λ n max u 2 , where (∆ n u) k = ∆ n u k , k ∈ Z. Proof. By (T 3 ), there exist constants ξ > 0 and β ∈ By using Hölder inequality and the fact that (a + b)   By the proof of Lemma 3.3, it is clear that Therefore, max , thus it can be seen that u (i) is a bounded sequence on E mT . As a result, u (i) possesses a convergence subsequence on E mT . Thus the P.S. condition is verified.
4. Proof of the main results. In this section, we shall prove Theorem 1.1 by using Linking Theorem.
In the remaining of the proof, we shall use the conclusion of Lemma 3.1, so as to obtain another nontrivial mT -periodic solution of (1.1) different fromū.
We have known that J satisfies the P.S. condition on E mT . Then, we shall verify the condition (J 2 ).
Take e ∈ ∂B 1 ∩ V . For any z ∈ W and s ∈ R, we denote u = se + z. Then Noting that for all u ∈ W , mT k=1 r k 1 + (∆ n u k ) 2 − 1 = 0, we have F (k, u k , u k−1 , · · · , u k−τ ) 0. Choosing h = id, we have sup u∈Q J(u) = c 0 . Since the choice of e ∈ ∂B 1 ∩V is arbitrary, we can take −e ∈ ∂B 1 ∩ V . Similarly, there exists a positive number R 2 > δ so that J(u) 0 for any u ∈ ∂Q 1 , where Using Linking Theorem once again, J possesses a critical value c σ > 0, where We claim that c = c 0 . Otherwise, suppose that c = c 0 , then sup u∈Q1 J(u) = c 0 .

PENG MEI, ZHAN ZHOU AND GENGHONG LIN
(P 4 ) There exist constants ρ 2 > 0 and p > 1 such that for k ∈ Z and τ j=0 v 2 j ρ 2 2 , Then for any given positive integer m, (1.1) possesses at least two nontrivial mTperiodic solutions.

5.
Example. As an application of Theorem 1.1, this section elaborates the example mainly. It is easy to verify all the assumptions of Theorem 1.1 are satisfied. Consequently, for any given positive integer m, (5.1) has at least three mT -periodic solutions.