In this article, by using critical point theory, we prove the existence of multiple -periodic solutions for difference equations with the mean curvature operator:
where is the set of integers. As a -periodic problem, it does not require the nonlinear term is unbounded or bounded, and thus, our results are supplements to some well-known periodic problems. Finally, we give one example to illustrate our main results.
Let and be the sets of integers and real numbers, respectively. For , denotes the discrete interval if .
In this article, we consider the following nonlinear difference equations with mean curvature operator
where is a positive real parameter, is the forward difference operator defined by , is an integer, and is -periodic function, satisfies for each , , and is the mean curvature operator defined by , where . For general background on the mean curvature operator, we refer to [1,2, 3,4].
Difference equations have been widely used in various research fields such as computer science, economics, biology, and other fields [5,6,7]. Recently, many excellent results for difference equations have been achieved, for example, positive solutions [8,9,10, 11,12], homoclinic solutions [13,14], and ground-state solutions . To study the existence of solutions for the discrete boundary value problems of difference equations, many authors give some important tools, such as the fixed point theory, critical point theory, and upper and lower solution techniques [3,8,16,17].
Problem (1) may be regarded as the discrete analog of the following one-dimensional prescribed mean curvature equation:
If , for and , then problem (1) is degenerated to
In addition, if and , many authors considered the existence results for the following difference equation by means of the variational methods,
Assume that is a positive integer. If , for , Zhou and Ling  first used the critical point theory to study the following discrete boundary value problem:
They proved the existence of infinitely many positive solutions of problem (5) under the suitable oscillating behavior of the nonlinear term at infinity.
In this article, our main aim is to use the critical point theory to establish the existence of multiple -periodic solutions of problem (1). We consider that problem (1) is a -periodic problem, and hence, problem (1) reduces to the following periodic boundary value problem:
Obviously, problem (6) is a more general difference equation with mean curvature operator. Although many excellent results have been worked out on the existence of periodic solutions for difference equations [18,19,20, 21,23], the multiple periodic solutions of the discrete boundary value problem involving the mean curvature operator show that no similar results were obtained in the literature.
This article is organized as follows. In Section 2, we present some definitions and results of the critical point theory. We establish the variational framework of problem (6) and transfer the existence of periodic solutions of problem (6) into the existence of critical points of the corresponding functional. In Section 3, we obtain the existence and multiple periodic solutions for boundary value problem (6), and we give an example to illustrate our main results.
In this section, our aim is to establish the existence of multiple periodic solutions of problem (6) by means of critical point theory. First, we recall some basic definitions and known results from the critical point theory.
Consider the -dimensional Banach space:
endowed with the norm
Let be a real Banach space and . A sequence is called a Palais-Smale sequence (P.S. sequence) for if is bounded and as . We say satisfies the Palais-Smale condition (P.S. condition) if any P.S. sequence for possesses a convergent subsequence in .
Let be a finite dimensional real Banach space and be a function satisfying the following structure hypothesis:
Assume that is a real positive parameter. for all , where , is coercive, that is, .
The following lemma will be used to prove our main results.
 Assume that and the following conditions hold,
For each , the functional satisfies the P.S. condition and it is bounded from below;
There exists such that .
Then, for , has at least three critical points.
For every , put
where , , then . By using , we can compute the Frećhet derivative as follows:
for all . It is clear that the critical points of are the solutions of problem (6).
(i) There exists a positive constant such that
Then satisfies the P.S. condition, and it is coercive on S.
For any sequence , with is bounded and as , there exists a positive constant such that . We shall prove the sequence is bounded.
If not, we assume as . From the condition , we take , there exists such that
where . Thus, we have
This contradicts the fact . Thus, the sequence is bounded in and the Bolzano-Weierstrass theorem implies that has a convergent subsequence.
Then is coercive on . The proof is complete.□
Since be a finite dimensional real Banach space, if is coercive on , then the conclusion of Lemma 2.1 holds. From the condition , the nonlinear term can be unbounded or bounded, and it does not require any asymptotic condition or a superlinear growth at infinity. It is different from the conditions of the literature .
3 Main results
Assume that the condition is satisfied and there exist two positive constants c and d with such that
Then, for each , problem (6) admits at least three periodic solutions.
Our aim is to apply Lemma 2.1 to prove our conclusion.
We take and defined as in (7) on the space . It is easy to verify that and satisfy the assumptions required in . In addition, it follows from Lemma 2.2 that of Lemma 2.1 hold. It remains to verify .
Let , we have
If , then we have
for each .
By (11), we obtain
Let for every , clearly, . Since , we have .
Hence, we obtain
By (10), we have . The proof is complete.□
If is a continuous odd function and the conditions of Theorem 3.1 hold. Then problem (6) admits at least five periodic solutions. In fact, let be one solution of problem (6). We see that is also solution of problem (6) when is odd. If the conditions of Theorem 3.1 are satisfied and is the trivial solution of problem (6), then problem (6) admits at least four different nontrivial periodic solutions and one zero solution. Moreover, if is not the solution of problem (6), then problem (6) admits at least six different nontrivial periodic solutions.
If the conditions holds and there exist two positive constants c and d with such that
Then, for each , problem (6) admits at least three periodic solutions.
In fact, in view of and (13) holds, we obtain
and hence, hypotheses of Theorem 3.1 are satisfied and our corollary holds.□
Let , assume that there exist two positive constants c and d with and the condition holds; moreover, we suppose that
for each ;
We note that for every by , let be the above, then it follows from that . Since , we have , which implies . By choosing as Theorem 3.1, we obtain
Thus, we have . By Lemma 2.1, we know that problem (6) admits at least two nontrivial periodic solutions for each . The proof is complete.□
Finally, we give an example to illustrate our results.
We consider the periodic problem (6) with
for all , is a given positive integer, obviously, . Then
For each , there exists such that
In addition, let , , , , , and , we have . Thus,
All conditions of Theorem 3.1 hold, we note that , and problem (6) admits at least three nontrivial periodic solutions for each .
The authors wish to thank the editor and the anonymous reviewers for their valuable comments and suggestions.
Funding information: This work was supported by the Scientific Research Project of Lüliang City (Grant No. Rc2020213) and Key Scientific Research Projects of Colleges and Universities in Henan Province (Grant Nos. 19B110009 and 20B110008).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.
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