The Existence of Multiple Periodic Solutions to a Class of Fourth-Order Difference Equations

Wu, Haoxin; Tan, Zixian; Hu, Xinfei; Xiao, Huafeng

doi:https://doi.org/10.1155/2022/7694885

Mathematical Problems in Engineering

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

Research Article | Open Access

Volume 2022 | Article ID 7694885 | https://doi.org/10.1155/2022/7694885

The Existence of Multiple Periodic Solutions to a Class of Fourth-Order Difference Equations

Haoxin Wu,¹Zixian Tan,¹Xinfei Hu,¹and Huafeng Xiao^1,2

Academic Editor: Maria Patrizia Pera

Received30 Jun 2022

Accepted01 Aug 2022

Published05 Sept 2022

Abstract

In this paper, we study the multiplicity of periodic solutions for a class of fourth-order difference equations. We estimate a lower bound on the number of periodic solutions when the nonlinearity is asymptotic both at the origin and at infinity. Our results generalize the reference’ results.

1. Introduction

On one hand, as we well know, many differential equations cannot be solved directly. In order to obtain approximate solutions iteratively by computer, we need to differentiate differential equations. For example, Gupta (cf. [1]) studied the bending of an elastic beam with simply-supported ends under an external force is described by the boundary-value problem

If we discretize the first equation of (1), we can obtain the following fourth-order difference equation

On the other hand, difference equations have been widely used as mathematical models to describe real-life situations in probability theory, matrix theory, electrical circuit analysis, combinatorial analysis, number theory, psychology and sociology, etc. For example, in 2015, Tariboon et al. (cf. [2]) introduced the quantum difference operators in orthogonal polynomials, basic hypergeometric functions, combinatorics, the calculus of variations, mechanics, and the theory of relativity. So it is worthwhile to explore this topic.

Many scholars studied the qualitative properties of difference equations such as disconjugacy, stability, attractivity, oscillation, bifurcation, and boundary value problems (cf. [3, 4]). In 2003, Guo and Yu firstly built the variational structure for difference equations (cf. [5–7]). Then they converted the existence of periodic solutions of difference equations to that of critical points of the variational functional. Since then, the study of solutions to discrete dynamical systems has attached the attention of many researchers, for example, boundary value problems(cf. [8, 9]), periodic solutions, subharmonic solutions(cf. [5–7, 10–12]), positive solutions(cf. [9]), solutions with the minimal period(cf. [12, 13]), homoclinic orbits(cf. [14]), and heteroclinic orbits(cf. [15]).

Focusing on fourth-order difference equations, the results are rare. To introduce the results, we give some notations. Let be the set of all natural numbers, integers, and real numbers, respectively. For , define when . In 2005, Cai, Yu, and Guo (cf. [16]) studied the fourth-order difference equationwhere such that for some positive integer . When was superlinear, using the linking theorem, some sufficient conditions were obtained for (3) to guarantee the existence and multiplicity of -periodic solutions.

In 2007, Zhao (cf. [17]) used the linking theorem to study the existence of solutions to the fourth-order difference equation

In his master’s thesis, he also gave some sufficient conditions to ensure the existence of homoclinic orbits of the following fourth-order difference equation

In 2008, Sun and Zhou studied a high-dimensional fourth-order difference system

Using the critical point theory, they (cf. [11, 18]) proved the existence and multiplicity of periodic solutions of (6) when satisfies super-quadratic conditions, sub-quadratic conditions, and asymptotically quadratic conditions, respectively.

In 2010, Cabada and Dimitrov studied the following fourth-order difference equation

Using Krasnoselskii’s fixed point theorem, the authors (cf. [19]) obtained some sufficient conditions to guarantee the existence and nonexistence of periodic solutions.

In 2011, Zhang studied the fourth-order difference equation with boundary value problems as follows

Using the linking theorem, the author (cf. [20]) proved the existence and multiplicity of periodic solutions to (8) with super-quadratic and sub-quadratic nonlinearity, respectively.

In 2020, Han and Ye (cf. [21]) studied the fourth-order nonlinear difference equation with periodic boundary conditions of the formthey applied the continuation theorem of Mawhin to ensure the existence of a nontrivial positive periodic solution. Also, they gave some sufficient conditions to ensure that (9) has no positive periodic solutions.

There are a few conclusions about the estimation of the number of periodic solutions in the literature. In 2006, Guo and Yu (cf. [10]) studied the second-order difference equationwhen is super-linear, using the geometrical index theory, the authors proved that (10) exists at least distinct -orbits of solutions with the minimal period . They also gave an example to illustrate that is the best lower bound.

In 2008, He and Chen investigated the multiplicity of periodic solutions for the following non-autonomous equation:

Using the critical point theory and Clark theorem, they (cf. [22]) gave some lower bound of periodic solutions to (11).

In 2013, Graef, Kong, and Wang studied the discrete fourth-order periodic boundary value problem

Using a variant of the Liusternik-Schnirelman theory, the authors (cf. [23]) obtained some sufficient conditions to guarantee the existence of multiple periodic solutions. Later, Dhar and Kong (cf. [24]) generalized the results to the following difference equation

More results from this direction, we refer to [25].

In this paper, we will study a class of fourth-order difference system when the nonlinearity is asymptotical both at the origin and at infinity. The rest of the paper is divided into three parts. In Section 2, we will study a class of a low dimensional fourth-order difference equation and estimate the lower bound of the number of periodic solutions. In Section 3, we will study a class of a high dimensional fourth-order difference system and find the lower bound of the number of periodic solutions. In Section 4, we will give three examples to illustrate our results.

2. A Low-Dimensional Fourth-Order Difference Equation

Consider the nonlinear fourth-order difference equationwhere is the forward difference operator defined by , .

Assume that: (F1) , and is odd with respect to variable . (F2) There exist , such that, , for all . (F3) There exist two functions such that as for all ; as for all .

Define

Let be the space of sequences , i.e.

For any , definethen is a vector space. Define the subspace of as follows

Define the norm and inner product on by

Define a linear map bywhere denotes the transpose of a vector. Then , defined in (19), is a linear homeomorphism with and is a Hilbert space, which can be identified with .

The functional on the Hilbert space iswhere .

Arguing similar as reference [6], we can prove the following result.

Lemma 1. Assume that satisfies (F1)–(F3). Then is continuously differentiable on . Critical points of are periodic solutions to equation (14).
Due to the identification of with , we write as . For convenience, is written aswhereIn order to compute the eigenvalues of , we introduce matrix as followsBy direct computation, the eigenvalues of can be given bySince , thenwhere . Obviously, 0 is an eigenvalue of with eigenvector . Other eigenvalues of are positive. When is odd, for any , is the eigenvalue of with multiplicity 2 and and the maximal eigenvalue is . When is even, for any , is the eigenvalue of with multiplicity 2. Furthermore, is the maximal eigenvalue of and the eigenvector corresponding to 16 is . For any , we denoteandIt is easy to see that and are the eigenvalues of corresponding to . Define and . By straightforward computation, one obtainsBy (F3), integrating both side of the two equalities, we getFor , set

Proposition 1. .
This proposition is obvious.
Denote by the diagonal matrix with diagonal elements . Set , , i.e.,where . Then (21) can be rewritten asTo apply critical point theory to prove our main results, we shall state some basic notations and lemmas, which can be found in reference [26].
Let be a real Hilbert space, be a real-valued function on . The norm and inner product of are given by and , respectively. We shall assume that and that satisfies the following Palais-Smale type condition (for short, (PS)-condition):(a)If a sequence is such that , is bounded below, and , then contains a convergent subsequence. For verification of Condition (A) in particular cases, it will be convenient to verify the following two conditions, which taken together are equivalent to the condition(A):(b)For every there exists and such that for such that and ;(c)If a bounded sequence is such that is bounded below, and , then contains a convergent subsequence.A function is called a quadratic form if there exists a symmetric bilinear form such that . A quadratic form is continuous if and only if there is a constant such that . In this case, there is a continuous symmetric linear operator from to the space dual to such that and is differentiable to all orders, the first and second derivatives being given by , . We shall say that is regular if for some . The index is defined to be the maximal dimension of a subspace of on which is negative.

Lemma 2. Let be a real Hilbert space and be real valued, functional on . Suppose further(1) and is an even, i.e., for all .(2) satisfies Condition (C).(3) is bounded below on bounded sets.(4), as , where is a quadratic form of index .(5), where is a continuous regular quadratic form of finite index and whereandas.Then for each integer such that , there is at least one antipodal pair of critical points of such that .

Lemma 3. Assume that (F1)–(F3) hold. Then satisfies Conditions (1)–(3) of Lemma 2.

Proof. Obviously, . Since is odd in , then is even in . Thus (1) of Lemma 2 holds.
Let be a bounded sequence such that is bounded below and . Since is finite-dimensional, then contains a convergence subsequence. Thus satisfies Condition (C).
Let be a bounded set. Since the minimal eigenvalue of is 0, then . Since is continuous, so is , which implies that is bounded for . Subsequently, is bounded below on bounded sets. Thus (1) of Lemma 2 holds.
Now let’s state and prove the main results.

Theorem 1. Suppose that satisfies (F1)–(F3). If there exist such that , then (14) possesses at least pairs of nonzero distinct -periodic solutions.

Proof. According to Lemmas 1 and 2, we need to check conditions (4)-(5) of Lemma 2.
By Proposition 1 and (32), and as . If , then there are eigenvalues of smaller than , which are , . Define by the negative eigenspace of . Then . Hence the index of is when .
By Proposition 1 and (32), and as . If , then there are eigenvalues of smaller than , which are , , . Define by the negative eigenspace of . Then . Hence index of is when . Using Lemma 2, has pairs difference nonzero critical points. Thus (14) has pairs of nonzero distinct -periodic solutions.

Corollary 1. Suppose that satisfies (F1)–(F3). If there exist such that , then (14) possesses at least pairs of nonzero distinct -periodic solutions.

Proof. When , the negative space of is as . Then the index of is as . The index of is when . Applying Theorem 1, (14) has pairs of nonzero distinct -periodic solutions.

Corollary 2. Suppose that satisfies (F1)–(F3). If , then (14) possesses at least non-zero distinct solutions.

Proof. If , then for all . Thus index of is 0 when . The index of is when . Applying Theorem 1, (14) possesses pairs nonzero distinct -periodic solutions.

Corollary 3. Suppose that satisfies (F1)–(F3). If , then (14) possesses at least pairs of nonzero distinct -periodic solutions.

Proof. When , the negative space of is as . Then the index of is . When , then for all . Thus index of is 0 when . Applying Theorem 1, (14) possesses pairs of nonzero distinct -periodic solutions.

3. A High-Dimensional Fourth-Order Difference System

In this section, we consider the nonlinear fourth-order difference system

Assume that: (G1) , and is odd with respect to variable . (G2) There exist and such that and , for all . (G3) There exist two matrix functions such that as for all ; as for all ,where denotes the general linear group of real matrices.

Let be the space of vector sequences , i.e.

For any , definethen is a vector space. Define the subspace of as follows

Define the norm and inner product on given bywhere and denote the usual norm and inner product in . Define a linear map bythen , defined in (38), is a linear homeomorphism with and is a Hilbert space, which can be identified with .

The functional , defined on the Hilbert space , is

Arguing similar as reference [6], we can prove the following result.

Lemma 4. Assume that satisfies (G1)–(G3). Then is continuously differentiable on . Critical points of are periodic solutions to (33).
Due to the identification of with , we write as . For convenience, is written aswherewhere is the identity matrix. By (G2) and (G3), we haveFor , set

Proposition 2. .

Proof. If , then for all . It follows from (42) and (44) that . For any , there exists such that for . If , then for all . ThusThe arbitrary of implies that . The other conclusion can be proved similarly.
For , define , where the matrices is given as follows(40) can be rewritten as can be decomposed as followswhere (respectively , ) denote the subspaces of on which is positive definite (respectively negative definite, null), . Setwhere denotes the dimension of the space .

Definition 1. A index of and is defined as followswhere (respectively denotes the subspace of on which is positive definite (respectively negative definite, null).
In order to apply critical point theory to prove our main results, we need the following lemmas.

Lemma 5. [27] Let be two constants. Let be a real Hilbert space and be a functional which satisfies the following assumption:(1), where is a bounded selfadjoint operator. is a functional whose Frechet derivative is a compact operator.(2).(3)Any sequence such that and as has a convergent subsequence.(4)There are two closed subspaces of , and , and a constant such thatThen the number of pairs of nontrivial critical points of is greater or equal to . Moreover, the corresponding critical values belong to .
(a) for ,(b) for .

Lemma 6. Assume that (G1)–(G3) hold. Then satisfies conditions (1), (2) and (4) of Lemma 5.

Proof. By Lemma 4, and . Observe that and . Since , which is a continuous function defined on a finite-dimensional space, thus is compact. Since is a symmetry matrix, thus is a bounded self-adjoint operator. Since is odd in , then is even in . Hence (1) and (2) of Lemma 5 hold.
Set , . There exist positive constants such thatIt follows that, for any ,Then there exist positive constants and such thatSetting , (4) (b) is satisfied.
By (G3), there exists such thatSince is continuous with respect to , is bounded for , whose boundedness is denoted by . Then is finite. Thusthen for every ,thus is bounded from below on . Setting with such that , then (4) (a) is satisfied. □
Now let’s state and prove the main results.

Theorem 2. Suppose that satisfies (G1)-(G3). If , then (3.1) possesses at least pairs of nonzero distinct solutions.

Proof. We need to check Conditions (3) of Lemma 5. Let be a sequence such that , , where such that . Now we show that is bounded. Suppose to the opposite, we can choose as . Clearly, can be written as . On one hand, sincethenOn the other hand,Obversing that , thenwhere satisfies for and for . Thuswhich contradicts with (58).
Applying Lemma 5, (33) possesses at least pairs of nonzero solutions if .
If , then . In this case, we replace by and let and . It is easy to check that Conditions (1)–(4) of Lemma 5 are satisfied. Applying Lemma 5, (33) possesses at least pairs of nonzero -periodic solutions.
In conclusion, (33) has at least pairs of nonzero solutions.

Theorem 3. Suppose that satisfies (G1)–(G3). Assume more, (G4) is bounded. as , uniformly for .then (33) possesses at least pairs of nonzero distinct -periodic solutions.

Proof. Let be a sequence such that , , where . Now we show that is bounded. Suppose to the contrary, we can choose as . Clearly, . Since , we getBy (G4), there exists such that . Then the above inequality implies thatthus is bounded. Arguing similarly, we prove is bounded. Thus, there exists such that . Subsequently, we haveTherefore is bounded from above. implies that is bounded. If holds, we can also prove that is bounded by modifying (3.11). Thus, is bounded and it contains a convergent subsequence.
Applying Lemma 5, (33) possesses at least pairs of nonzero -periodic solutions.

4. Examples

In this section, we will give three examples to illustrate our results. Consider the following difference equationwhere

It is easy to check that satisfies (F1)–(F3). Applying Theorem 1, (65) has 4-pairs nonzero distinct 15-periodic solutions since .

Next, we will consider the following fourth-order difference systemwhere is continuous and for all and . Let and large enough,

Set

Compute the eigenvalues of and . The space is split intowhere , and , are denoted by positive, negative eigenspaces of and respectively. Computing directly, we obtain

Applying Theorem 2, (67) possesses 14 pairs of 8-periodic solutions.

Consider fourth-order difference system (67), where is give below

Set

Compute the eigenvalue of and . The space is split intowhere , , and , , are denoted by positive, negative eigenspaces, and null space of and , respectively. Computing directly, we obtain

Applying Theorem 3, (67) possesses 5 pairs of 8-periodic solutions.

Data Availability

No data is available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

This project is supported by National Natural Science Foundation of China (No. 11871171).

References

C. P. Gupta, “Existence and uniqueness results for the bending of an elastic beam equation at resonance,” Journal of Mathematical Analysis and Applications, vol. 135, pp. 209–225, 1988.
View at: Google Scholar
J. Tariboon, S. Ntouyas, and P. Agarwal, “New Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional Q-Difference Equations,” Adv. Difference Equ, vol. 2015, p. 18, 2015.
View at: Google Scholar
S. Chen, “Disconjugacy, disfocality, and oscillation of second order difference equations,” Journal of Differential Equations, vol. 107, pp. 383–394, 1994.
View at: Google Scholar
P. Hartman, “Difference equations: disconjugacy, principal solutions, Greens functions, complete monotonicity,” Transactions of the American Mathematical Society, vol. 246, pp. 1–30, 1978.
View at: Google Scholar
Z. M. Guo and J. S. Yu, “Existence of periodic and subharmonic solutions for second-order superlinear difference equations,” Science in China, Series A, vol. 46, pp. 506–515, 2003.
View at: Google Scholar
Z. Guo and J. Yu, “Existence of periodic and subharmonic solutions to subquadratic second-order equations,” Journal of the London Mathematical Society, vol. 68, pp. 419–430, 2003.
View at: Google Scholar
Z. Guo and J. Yu, “Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems,” Nonlinear Anal, vol. 55, pp. 969–983, 2003.
View at: Google Scholar
J. Yu and Z. Guo, “On boundary value problems for a discrete generalized Emden-Fowler equation,” Journal of Differential Equations, vol. 231, pp. 18–31, 2006.
View at: Google Scholar
J. Yu, B. Zhu, and Z. Guo, “Positive solutions for multiparameter semipositone discrete boundary value problems via variational method,” Advances in Differential Equations, vol. 2008, p. 15, 2008.
View at: Google Scholar
Z. Guo and J. Yu, “Multiplicity results for periodic solutions to second order difference equations,” Journal of Dynamics and Differential Equations, vol. 18, pp. 943–960, 2006.
View at: Google Scholar
Q. Sun and Z. Zhou, “Existence of multiplicity periodic solutions to discrete Hamiltonian systems,” Math. Appl., vol. 21, pp. 529–534, 2008.
View at: Google Scholar
J. Yu, Y. Long, and Z. Guo, “Subharmonic solutions with prescribed minimal period of a discrete forced pendulum equation,” Journal of Dynamics and Differential Equations, vol. 16, pp. 575–586, 2004.
View at: Google Scholar
J. Yu, “Subharmonic solutions with prescribed minimal period of a class of nonautonomous Hamiltonian systems,” Journal of Dynamics and Differential Equations, vol. 20, pp. 787–796, 2008.
View at: Google Scholar
M. Ma and Z. Guo, “Homoclinic orbits for second order self-adjoint difference equations,” Journal of Mathematical Analysis and Applications, vol. 323, pp. 513–521, 2006.
View at: Google Scholar
H. Xiao and J. Yu, “Heteroclinic orbits for a discrete Pendulum equation,” Journal of Difference Equations and Applications, vol. 17, pp. 1267–1280, 2011.
View at: Google Scholar
X. Cai, J. Yu, and Z. Guo, “Existence of periodic solutions for fourth-order difference equations,” Computers & Mathematics with Applications, vol. 50, pp. 49–55, 2005.
View at: Google Scholar
D. Zhao, Periodic Solutions and Homoclinic Orbits to Fourth-Order Difference Equations, Kunming University of Science and Technology, vol. 1, 2007, Master thesis.
Q. Sun, The Existence of Periodic Solutions to Fourth-Order Nonlinear Difference Equations, Guangzhou University, 2008, Master thesis.
A. Cabada and N. Dimitrov, “Multiplicity results for nonlinear periodic fourth-order difference equations with parameter dependence and singularities,” Journal of Mathematical Analysis and Applications, vol. 371, pp. 518–533, 2010.
View at: Google Scholar
G. Zhang, Variational Methods for Fourth-Order Difference Boundary Value Problem and Second Order Difference System, Lanzhou University, 2011, Master thesis.
X. Han and F. Ye, Existence of Periodic Solutions for a Class of Fourth-Order Difference Equation, 2020.
View at: Publisher Site
T. He and W. Chen, “Multiplicity of periodic solutions of a class of nonlinear second-order difference equation (in Chinese),” Chinese Journal of Engineering Mathmatics, vol. 25, pp. 804–810, 2008.
View at: Google Scholar
J. Graef, L. Kong, and M. Wang, “Existence of multiple solutions to a discrete fourth-order periodic boundary value problem,” Journal of Difference Equations and Applications, vol. 1, pp. 291–299, 2013.
View at: Google Scholar
S. Dhar and L. Kong, “Existence of multiple solutions to a discrete fourth-order periodic boundary value problem via variational method,” Differ. Equ. Dyn. Syst., vol. 2, 2018.
View at: Publisher Site | Google Scholar
J. Graef, L. Kong, and X. Liu, “Existence of solutions to a discrete fourth-order periodic boundary value problem,” Journal of Difference Equations and Applications, vol. 22, pp. 1167–1183, 2016.
View at: Google Scholar
D. Clark, “A variant of the Lusternik-Schnirelman theory,” Indiana University Mathematics Journal, vol. 22, pp. 65–74, 1972.
View at: Google Scholar
V. Benci, “On critical point theory for indefinite functionals in the presence of symmetries,” Transactions of the American Mathematical Society, vol. 274, pp. 533–572, 1982.
View at: Google Scholar
H. Bin, J. Yu, and Z. Guo, “Nontrival periodic solutions for asymptotically linear resonant difference problem,” Journal of Mathematical Analysis and Applications, vol. 322, pp. 477–488, 2006.
View at: Google Scholar
H. Xiao and B. Zheng, “The existence of multiple periodic solutions of nonautonomous delay differential equations,” Journal of Applied Mathematics, vol. 2011, p. 15, 2011.
View at: Google Scholar

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