Periodic Solutions with Prescribed Minimal Period for <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2725pt" id="M1" height="11.2929pt" version="1.1" viewBox="-0.0657574 -11.0204 17.0458 11.2929" width="17.0458pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-51" d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"/></g><g transform="matrix(.017,0,0,-0.017,8.237,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g></svg> th-Order Nonlinear Discrete Systems

Shi, Haiping; Luo, Peifang; Huang, Zan

doi:https://doi.org/10.1155/2021/2432761

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 2432761 | https://doi.org/10.1155/2021/2432761

Periodic Solutions with Prescribed Minimal Period for th-Order Nonlinear Discrete Systems

Haiping Shi,¹Peifang Luo,¹and Zan Huang²

Academic Editor: Douglas R. Anderson

Received25 Jun 2021

Accepted13 Aug 2021

Published06 Sept 2021

Abstract

In this paper, by using the critical point theory, some new results of the existence of at least two nontrivial periodic solutions with prescribed minimal period to a class of th-order nonlinear discrete system are obtained. The main approach used in our paper is variational technique and the linking theorem. The problem is to solve the existence of periodic solutions with prescribed minimal period of th-order discrete systems.

1. Introduction

In the following and in the sequel, we denote by , , and the sets of all natural numbers, integers, and real numbers, respectively. The symbol is defined by the transpose of a vector. Let be the greatest integer function. For any integers and with , we let denote the discrete interval and .

Now, we concern with the following 2th-order nonlinear discrete system:where is the forward difference operator defined by , , and for , , and for a given integer .

We may think of (1) as a discrete analogue of the following th-order differential equation:

Equations similar in structure to (2) have been studied by many authors [1–6]. The difficulty of this paper comparing with system (2) is that there are few known techniques for studying the existence of periodic solutions with minimal period of (1).

Denote

Throughout this paper, we suppose that there is a function with , , , and

Bin [7] in 2013 considered the second-order discrete Hamiltonian systems:

By using Morse theory, some new results concerning the existence of nontrivial periodic solution are obtained.

Using the critical point method, Liu et al. [8] studied the following forward and backward difference equation:

Some new criteria for the existence and multiplicity of periodic and subharmonic solutions are established.

In 2016, Leng [9] established some new criteria for the existence and multiplicity of periodic and subharmonic solutions to the th-order difference equation with -Laplacianusing the linking theorem in combination with variational technique.

By establishing a new proper variational framework and using the critical point theory, He [10] established some new existence criteria to guarantee that the th-order nonlinear difference equation containing both advance and retardation with -Laplacianhas infinitely many homoclinic orbits.

Lin and Zhou [11] concerned with the existence and multiplicity of periodic and subharmonic solutions of the following th-order difference equation

By using the critical point theory, some new sufficient conditions are obtained and some previous results are generalized.

Xia [12] in 2018 considered the existence of periodic solutions for a higher order nonlinear difference equationby making use of the critical point theory. Some existence criteria are established. Some results are generalized.

In 1978, Rabinowitz [13] proposed a conjecture that the Hamiltonian system has a nonconstant periodic solution with prescribed minimal period under some given conditions. From then on, there has been much progress [14–16] on Rabinowitz’s conjecture under various conditions. Ambrosetti and Mancini [14] assumed that the dual functional is bounded from below and the Hamiltonian system has a minimum to which correspond a solution with given minimal period. Ekeland and Hofer [16] proved that if the Hamiltonian system is flat near an equilibrium and superquadratic near infinity, it has a periodic solution with minimal period. The estimate of number of periodic solutions was established in [15]. In contrast to differential equations, the research on periodic solutions with prescribed minimal period of higher order discrete systems is fresh and there are very few literature (see [7–12, 17–29]) on it. Comparing this paper with references [8–11], the advantages and differences of this paper are that the existence of periodic solutions with prescribed minimal period of (1) is obtained in this paper; however, only periodic solutions are obtained in the references [8–11]. Given integer , Long [23] considered the following -cycle discrete Hamiltonian systems:

By making use of minimax theory and geometrical index theory, some results on the existence and multiplicity of subharmonic solutions with prescribed minimal period to the abovementioned discrete Hamiltonian systems are obtained. Yu et al. [27] in 2004 obtained some sufficient conditions on the existence of subharmonic solutions with prescribed minimal period for the second-order difference equation by using variational methods. Therefore, there is still spacious room to explore the periodic solutions with prescribed minimal period of higher order discrete systems. Motivated by the papers [9, 12], for a given integer with , the aim of this paper is to obtain some new results for the existence of at least two nontrivial periodic solutions with minimal period to a th-order discrete system by using critical point method.

Here, we give the existence results of at least two nontrivial periodic solutions with minimal period as follows.

Theorem 1. Suppose that satisfies the following assumptions: There are constants and such that There are constants and such that There are three constants and such that If is a rational number, is a solution of (1) with a minimal period , and also has a minimal period , then must be an integer. Denote by the least prime factor of : Then, (1) possesses at least two nontrivial periodic solutions with minimal period .

Corollary 1. Suppose that satisfies andIfthen there is such that for any prime integer , (1) possesses at least two nontrivial periodic solutions with minimal period .

Theorem 2. Suppose that satisfies and the following assumptions: uniformly for . There are constants and such that Then, (1) has at least two nontrivial periodic solutions with minimal period .

Theorem 3. Suppose that satisfies and the following assumptions. Ifthen (1) has at least two nontrivial periodic solutions with minimal period .

Corollary 2. Suppose that satisfies and the following assumptions. Ifthen there is such that for any prime integer , (1) possesses at least two nontrivial periodic solutions with minimal period .

The remainder of this paper is organized as follows. In Section 2, we build the variational functional and gather some basic notations that are necessary in the proofs of our main theorems. In Section 3, we state some useful lemmas. In Section 4, the main results will be proved.

Regarding the basis for variational methods, we refer the reader to [30]. Regarding the basic knowledge of integral inequalities and extended hypergeometric functions, the reader is referred to [31].

2. Preliminaries

In this section, we shall establish the variational framework associated with (1) and gather some basic notations that are necessary in the proofs of our main theorems.

Let the vector space be defined byand for any , define the inner productand the norm

For any , let be the functional defined by

Then, is continuously differentiable and

Thus, is a critical point of on if and only if

Therefore, we reduce the problem of finding -periodic solutions of (1) to that of seeking critical points of the functional on .

Denote

It is clear that the eigenvalues of are

Furthermore, is positively semidefinite and all of eigenvalues of are positive except for 0, and

Obviously, 0 is an eigenvalue of and is an eigenvector associated to 0. Let , then is an invariant subspace of . Denote by .

The eigenvectors of corresponding to are defined by

Set

If is odd, then . For any and ,where , and are constants.

If is even, then 4 is the eigenvalue of . Let denote the eigenvector corresponding to 4, and span . We have . For any and ,where , and are constants.

Suppose that is a real Banach space and . As usual, is said to satisfy the Palais–Smale condition if every sequence , such that is bounded and , has a convergent subsequence. The sequence is called a Palais–Smale sequence.

3. Some Lemmas

To apply critical point theory to study the existence of periodic solutions with minimal period of (1), some lemmas should be stated in this section which will be used in proofs of our main results.

Below, we denote by the open ball centered at with radius , as its closure, and as its boundary.

Lemma 1 (linking theorem [30]). Let be a real Banach space, , where is finite dimensional. Suppose that satisfies the Palais–Smale condition and the following: There are positive constants and such that There is and a positive constant such that , where Then, possesses a critical value , whereand , where denotes the identity operator.
SetWe have , then

Lemma 2. Suppose that satisfies . Then,is bounded from above in .

Proof. For any , by , we havewhere . SincethenThe proof is finished.

Lemma 3. Suppose that satisfies . Then,satisfies the Palais–Smale condition.

Proof. Assume that is a bounded sequence from the lower bound. Then, there is a positive constant such thatThe proof of Lemma 2 implies thatTherefore,It comes from that we can find a positive constant such that for any , . As a consequence of this, we know that the sequence is a bounded in the finite-dimensional space . Thus, it has a convergent subsequence. The Palais–Smale condition is verified.

Lemma 4. Suppose that is a critical point of on . Then, is a critical point of on .

The proof of Lemma 4 is similar to the proof of Lemma 2.2 in [12]. For simplicity, we omit its proof.

Let

Lemma 5. Suppose that satisfies and . If is a critical point of on , then has a minimal period .

Proof. Suppose, for the sake of contradiction, that exists a minimal period . In view of the condition , we have .
Similarly, can be written in the form ofThus,where . It is obvious thatTherefore, by ,That is, which is a contradiction with the condition , and the proof of Lemma 5 is now complete.

4. Proofs of the Main Theorems

In this section, the proofs of Theorems 1–3 and Corollaries 1 and 2 are given by using the critical point theory.

Proof. of Theorem 1. In view of Lemma 2, is bounded from above in .
SetThus, there is a sequence on such thatIn addition, by the proof of Lemma 2, for any ,Therefore, . This means that is bounded. Consequently, has a convergent subsequence. We define it as . DenoteOn account of the continuity of in , there must be a point , . Obviously, is a critical point of .
For any , from the condition ,where . It is easy to see thatTherefore,Take . Thus, for any ,Consequently,Furthermore, there are positive constants and such that for any ,For any , note that , we haveThus, and the critical point of corresponding to the critical value is a nontrivial periodic solution of (1) with period .
Choose . DenoteWe havewhere . SincethenAccordingly, there exists a positive number such thatwhere . Applying the linking theorem, has a critical value , whereand .
Similar to the Proof of Theorem 1 in [22], we can prove that (1) has at least two -periodic nontrivial solutions and so we omit it. By Lemma 5, it suffices to prove thatFrom ,Thus,ChooseIt comes from and thatwhere is a constant. In addition,Consequently,It is easy to see thatThus,The Proof of Theorem 1 is completed.

Proof. of Corollary 1. Since is a prime integer and , it is easy to see that . Therefore,In virtue of Theorem 1, the conclusion of Corollary 1 is obtained. The proof of Corollary 1 is fulfilled.

Proof of Theorem 2. In fact, it is evident that condition implies and condition implies . As a result of Theorem 1, Theorem 2 holds. The result of Theorem 2 is achieved.

Remark 1. Similar to the proof of Theorem 1, we can also prove that Theorem 3 is right. For simplicity, we omit its proof. Thanks to Theorem 3, the conclusion of Corollary 2 is obviously accomplished.

Remark 2. A real example is given by using the main results of this paper.
For , assume thatWe haveIt is easy to verify that all the assumptions of Theorem 1 are satisfied. Consequently, (77) possesses at least two nontrivial periodic solutions with minimal period (20).

Data Availability

This paper is purely theoretical, so there are no supporting data.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Authors’ Contributions

All authors contributed equally and significantly to writing this manuscript. All authors read and approved the final manuscript.

Acknowledgments

This project was supported by the “Innovation and Strengthen University” Project of Guangzhou Maritime University (D234, F320519, A330106, B510647, F321430, and F321108).

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Copyright © 2021 Haiping Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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