Abstract
We consider a 2mth-order nonlinear difference equation containing both advance and retardation with -Laplacian. Using the critical point theory, some new and concrete criteria for the existence of homoclinic solutions with mixed nonlinearities are obtained.
1. Introduction
Consider the following 2mth-order nonlinear difference equation containing both advance and retardation with -Laplacian:where is the forward difference operator defined by and for , is real valued for each , , and are T-periodic in n for a given positive integer T, and is a special ϕ-Laplacian operator [1] defined by .
It is well known that is the mean curvature operator given by the corresponding potential . And it is such thatwhere may be seen as the acceleration in classical relativity. Actually, it is important to determine the existence and regularity properties of maximal and constant mean curvature hypersurfaces, as they provide Riemannian submanifolds that reflect spatiotemporal properties. There have been extensive study and application of nonlinear difference equations with ϕ-Laplacian. For example, scalar equations were considered in [2, 3] using topological methods, and T-periodic solutions have been studied in [1] by variational methods.
Bonheure et al. [4] in 2007 studied classical and nonclassical solutions of a prescribed curvature equation:
This problem depends on the behavior at the origin and at infinity of the potential . Their method is essentially variational and is based on a regularization of the action functional associated with the curvature problem.
Due to their wide range of applications, there has been tremendous interest in studying some properties of the nonlinear discrete -Laplacian equations, including periodic solutions [1], boundary value problems [5, 6], and homoclinic orbits [7, 8]. For instance, using the mountain pass lemma, Zhou and Su [5] obtained some sufficient conditions for the first time on the existence of solutions of the 2nth-order -Laplacian difference equation with the boundary value conditions, and by the critical point theory, Zhou and Ling [6] obtained some sufficient conditions on the existence of infinitely many positive solutions of the boundary value problems for a second-order -Laplacian difference equation. The importance of nonlinear difference equations is described in [9, 10], with applications involving statistics, computing, electrical circuit analysis, dynamical systems, and economics. And they appear naturally as discrete analogue of differential equations which model various diverse phenomena.
We assume that for each , then is a solution of (1), which is called the trivial solution. As usual, we say that a solution of (1) is homoclinic (to 0) if . In addition, if , then u is called a nontrivial homoclinic solution.
In recent years, the existence and multiplicity of homoclinic solutions for various discrete systems have been investigated by many researchers. For some recent work, the reader can refer to [11–22] and the reference therein. For example, by using the symmetric mountain pass theorem, Chen and Tang [12] studied the existence of infinitely many homoclinic orbits for the fourth-order difference systems containing both advance and retardation:
Kong [20] applied the critical point theory to study the existence of at least three homoclinic solutions for a p-Laplacian difference equation of higher order with both advance and retardation:
However, to the best of our knowledge, there is no work on the existence of homoclinic solutions for -Laplacian difference equations containing both advance and retardation which have important background and significance in the field of cybernetics and biological mathematics [23, 24]. Motivated by the reasons aforementioned, in this paper, we will establish some new sufficient conditions for the existence of homoclinic solutions to (1) in a very general case of the nonlinear functional.
On the other hand, we notice that the nonlinear term f is only either superlinear or asymptotically linear at which plays an important role in the existence of homoclinic solutions when the similar arguments were considered in many references [25–28]. But in this paper, the nonlinearities can be mixed superlinear with asymptotically linear at , see Remark 1 for details. In fact, our conditions on the potential are rather relaxed, and some existing results in the literature are improved (see Remark 2). Moreover, we give an example to illustrate the main result.
Now, we establish the main result.
Theorem 1. Assume that the following conditions hold: there exists a functional with and it satisfies there exists a real sequence such that for . as .If for each , then (1) has at least a nontrivial solution u in .
Remark 1. The condition shows that the nonlinear terms mix superlinear nonlinearities with asymptotically linear ones at .
Remark 2. If and , Theorem 1 can reduce to Theorem 2 when ϕ-Laplacian is -Laplacian in [8]. And our sufficient conditions are based on the limit superior and limit inferior, which are more applicable.
This paper is organized as follows: In Section 2, we establish the variational framework associated with (1) and recall some related fundamental results for convenience. In Section 3, some lemmas are proved, and then we complete the proof of our results by the mountain pass lemma. Finally, we illustrate our main result with an example in Section 4.
2. Preliminaries
This section is to establish the corresponding variational framework for (1) and cite some basic conclusions for the coming discussion.
Letthen, the corresponding inner product denoted by in and the corresponding norm in is denoted by .
Let S be the set of all two-sided sequences, that is,
Then, S is a vector space with for . For any fixed positive integer k, we define the subspace of S as
Obviously, is isomorphic to , and hence can be equipped with the inner product and norm asrespectively. We also define a norm in by
Consider the functional in defined byand the Fréchet derivative is given by
Equation (15) implies that (1) is the corresponding Euler–Lagrange equation for . Therefore, we have reduced the problem of finding a nontrivial solution of (1) to that of seeking a nonzero critical point of the functional .
It is easy to see that the critical points of in are exactly -periodic solutions of system (1).
Let H be a Hilbert space, and denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on H. We conclude this section with some known results.
Definition 1. Let . A sequence is called a Cerami sequence ( sequence for short) for J, if for some and as . We say, J satisfies the Cerami condition ( condition for short) if any sequence for J possesses a convergent subsequence.
Let be the open ball in H with radius r and center 0, and let denote its boundary.
Lemma 1. (mountain pass lemma [29]). If and satisfies the following conditions: there exist and such that . Then, there exists a sequence for the mountain pass level c which is defined bywhere
3. Proof of Main Result
In order to prove Theorem 1, we need several lemmas.
Let
Lemma 2. Under the assumptions of Theorem 1, the functional satisfies the condition.
Proof 1. Let be a sequence for . We need to show that has a convergent subsequence. Since is finite dimensional, it suffices to show that is bounded. Since , then there exists such that and for . So, we have for . Then, by (14), (15), andwe haveFrom and , there exists such thatthen, (20) implies that for , that is,Since is finite dimensional, and are equivalent. Then, (22) implies that is bounded. The proof is completed.
Lemma 3. Under the assumptions of Theorem 1, there exists such that has at least a nonzero critical point in for each .
Proof 2. We first show that satisfies the conditions of Lemma 1. From , there exists such thatThen, for with ,Taking , then .
Since , there exists such thatLet satisfySince is a bounded semipositive self-adjoint linear operator in and , there exists with such that . Let be large enough such thatFor , define byBy (), there exists , such thatThen, for ,Thus,It can easily be seen that for each . Then, we have andNow that we have verified all assumptions of Lemma 1, and then we know possesses a sequence for the mountain pass level withwhereAccording to Lemma 2, has a convergent subsequence such that as for some . Since , we haveas . By the uniqueness of the limit, we obtain that is a critical point of corresponding to . Moreover, is nonzero as .
Lemma 4. There exist such thatholds for every critical point obtained by Lemma 3, of in with , where is defined in Lemma 3.
Proof 3. For and , we define as ; similarly to the derivation of [5], we can findLetthen,Obviously, is independent of k.
Since is a critical point of , by (14), (15), and (39), we haveFrom , there exists such thatBy combining (40), it implies that for each , that is,From (15), we haveBy , there exists such thatwhich combining with (43) givesArguing by a contradiction , we haveThe proof is complete.
With the help of the preceding lemmas we can prove Theorem 1.
Proof of Theorem 1. For every , has a critical point , which is obtained by Lemma 3. Moreover, there exists such thatNote thatBy the periodicity of and , we see that is also a solution of (48). Without the loss of generality, we may assume that in (47). Moreover, passing to a subsequence of , if necessary, we can also assume that for and some integer such that . It follows from (47) that we can choose a subsequence, still denoted by , such thatThen, is a nonzero sequence as (47) which implies . It remains to show that , and it is a solution of (1).
LetSince is continuously differentiable in the second and third variables and T-periodic in n, letit implies . Thus, for ,Then, combining (40), (43), and (45) gives usThus, we haveFor each , let , and it follows from (54) thatIt is clear that as and by the arbitrariness of ϱ.
Indeed, for each , letting in (48) gives usthat is, satisfies (1).
Consequently, we infer that is a nontrivial solution of (1). We have thus proved Theorem 1.
4. An Example
As an application of Theorem 1, we give an example to illustrate our main result.
Example 1. Consider the difference equation (1), withwhere and T is a given positive integer. Then,We can see thatdue to , it follows thatAnd we knowThen, it implies thatIt is easy to verify that all 1.1 are satisfied. Consequently, (1) has at least a nontrivial solution u in .
5. Conclusion
In this work, we have considered a higher order nonlinear difference equation containing both advance and retardation with -Laplacian. To the best of our knowledge, this is the first time to discuss the existence of homoclinic solutions for such equations. Applying the variational method and mountain pass theorem with Cerami’s condition, we have derived some new sufficient conditions for the existence of homoclinic solutions to (1) in a very general case of the nonlinear functional. By setting up the variational framework with (1), we have reduced the problem of finding a nontrivial solution of (1) to that of seeking a nonzero critical point of the functional . It is worth mentioning that the nonlinear term f can be mixed superlinear with asymptotically linear at . In fact, our conditions on the potential are rather relaxed, and some existing results in the literature are improved. Finally, we have given an example to illustrate the main result.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant no. 11571084) and the Program for Changjiang Scholars and Innovative Research Team in University (Grant no. IRT-16R16).