Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

Li, Shengjun; Zhang, Fang

doi:https://doi.org/10.1155/2021/9954156

Journal of Function Spaces

On this page

Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Special Issue

Unique and Non-Unique Fixed Points and their Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 9954156 | https://doi.org/10.1155/2021/9954156

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

Shengjun Li¹and Fang Zhang²

Academic Editor: Liliana Guran

Received31 Mar 2021

Revised28 May 2021

Accepted06 Jul 2021

Published13 Jul 2021

Abstract

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

1. Introduction

Difference systems are widely used in modeling real-life phenomena [1] and references therein. In this paper, we establish the existence positive solutions for the following nonlinear difference systems:with the boundary conditions:where , , , and , By a periodic solution, we mean a function , solving (1) and (2) and such that for all . We call boundary condition (2) the periodic boundary conditions which are important representatives of nonseparated boundary conditions. For convenience, we denote by , , and the sets of all integer numbers, natural numbers, and real numbers, respectively. For , let when . As usual, denotes the forward difference operator defined by

In particular, the nonlinearity may have a repulsive singularity at , from the physical explanation, which means that , uniformly in .

Such repulsive singularity appears in many problems of applications such as the Brillouin focusing systems and nonlinear elasticity [2].

System (1) can be viewed as a discretization of the following more general class of the Sturm singular second-order differential system:

Such systems, even in case , where they are referred to as being of Klein-Gordon or Schrödinger type, appear in many scientific areas including fluid mechanics, gas dynamics, and quantum field theory. During the last few decades, the study of the existence of periodic solutions for singular differential equations has deserved the attention of many researchers [3–11]. Tracing back to 1987, Lazer and Solimini [5] investigated the singular model:where are -periodic functions and the mean value of is negative, . One of the common conditions to guarantee the existence of positive periodic solution is a so-called strong force condition (corresponds to the case in (5)) [11, 12]. For example, if we consider the system:with ; the strong force condition holds for . On the other hand, the existence of positive periodic solutions of the singular differential equations has been established with a weak force condition (corresponds to the case in (5)) [13–15].

From then on, some classical tools have been used to study singular differential equations in the literature, including the degree theory [6, 11, 16], the method of the upper and lower solutions [8, 17], Schauder’s fixed point theorem [14], some fixed point theorems in cones for completely continuous operators [13, 18], and a nonlinear Leray-Schauder alternative principle [19].

For the existence of periodic solutions of difference equations, some results have been obtained using the variational methods or the topological methods [1, 20–25]. For example, by minimax principle, Guo and Yu [23] discussed the existence of periodic solutions for difference equation:where the nonlinearity is of superlinear or sublinear growth at infinity. Based on the method of the upper and lower solutions, Atici and Cabada [21] studied the existence of periodic solutions for difference equation:

In [26], Zhou and Liu investigated the following autonomous difference equations:

By Conley index theory, the author showed that the suitable assumptions of asymptotically linear nonlinear are enough to guarantee the existence of periodic solutions.

In this paper, we establish two different existence results of positive periodic solutions for (1) and (2) and proof of the existence of positive solutions; the first one is based on an application of a nonlinear alternative of Leray-Schauder, which has been used by many authors [19, 27, 28] and references therein; the second one is based on a fixed point theorem in cones. Our main motivation is to obtain new existence results for positive periodic solutions of the system:

Here, we emphasize that the new results are applicable to the case of a strong singularity as well as the case of a weak singularity and that does not need to be positive.

The rest of this paper is organized as follows. In Section 2, some preliminary results will be given. In Section 3, we will state and prove the main results. We will use the notation for each , for , we write , if . We say that a function is nondecreasing if for with . For a given function defined on , we denote its maximum and minimum by and , respectively.

2. Preliminaries

For , let us denote by and the solutions of the corresponding homogeneous equations:satisfying the initial conditions:

Let

Throughout this paper, we always assume that

(H) For each

Lemma 1 (see [29]). If (H) holds, then .

Lemma 2 (see [29]). Assume (H) holds. For the solution of the problem:the formulaholds, whereis the Green’s function; the number is defined by (13).

Lemma 3 (see [29]). Under condition (H), the Green’s function of the boundary value problem (14) is positive, i.e., for .
We denoteObviously, and .

Remark 4. If , then Green’s function of the boundary value problem (14) has the form:where . If is even, a direct calculation shows that

3. Main Results

In this section, we state and prove the new existence results for (1). In order to prove our main results, the following nonlinear alternative of Leray-Schauder is needed, which can be found in [30].

Lemma 5. Assume is a relatively compact subset of a convex set in a normed space X. Let be a compact map with Then, one of the following two conclusions holds:(i) has at least one fixed point in (ii)There exist and such that LetThen, is a Banach space with the normWe takewith the normDefinewhich corresponds to the unique solution of (14), and the operator by , where

Now, we present the first existence result of the positive solution to problem (1).

Theorem 6. Suppose that condition (H) holds and . Furthermore, we assume that
(H₁) For each constant , there exists a function for all such that each component of satisfies for all
(H₂) For each component of , there exist nonnegative functions , , and such thatand > 0 is nonincreasing and is nondecreasing in
(H₃) There exists a positive number r such that andfor all . Here,

Then, (1) and (2) has at least one positive periodic solution with for all and .

Proof. We first show thattogether with (2) has a positive solution satisfying for and . If this is true, it is easy to see that will be a positive solution of (1) and (2) with sinceSince (H₃) holds, let , we can choose such that andfor all .
Fix . Consider the family of systemswhere and for each ,Problem (29) and (2) are equivalent to the following fixed point problem:for each , here, we used the factWe claim that any fixed point of (34) for any must satisfy . Otherwise, assume that is a fixed point of (34) for some such that . Without loss of generality, we assume that for some .
Thus, we haveHence, for all , we haveTherefore,Using (34), we have from condition (H₂), for all ,Therefore,This is a contradiction to the choice of , and the claim is proved.
From this claim, the nonlinear alternative of Leray-Schauder guarantees thathas a fixed point, denoted by , in , i.e.,has a periodic solution with .
Next, we claim that these solutions have a uniform positive lower bound, that is, there exists a constant , independent of , such thatfor all . To see this, we know from (H₁) that there exists a continuous function such that each component of satisfies for all . Now, let be the unique solution towith (2), here . Then, we havefor each , hereNext, we show that (43) holds for . To see this, for each , since and , we haveThe fact and (43) show that for each , is a bounded family on . Moreover, we havewhich implies thatThus, the Arzela–Ascoli theorem guarantees that has a subsequence, converging uniformly on to a function . Let , satisfies for all and . Moreover, satisfies the integral equation:Letting , we arrive athere, we have used the fact that is with respect to with and satisfying . Therefore, is a positive periodic solution of (1) and satisfies

Corollary 7. Assume that (H) holds, , . Then, for each with , we have(i)if , then (10) has at least one positive periodic solution for each (ii)if , then (10) has at least one positive periodic solution for each , where is some positive constant

Proof. We will apply Theorem 6. To this end, assumption (H₁) is fulfilled by . If we takeand , then (H₂) is satisfied.
LetThen, the existence condition (H₃) becomesfor some . So, (10) has at least one positive periodic solution forNote that if and if . We have (i) and (ii).

In more general, we can obtain the following result.

Corollary 8. Assume that (H) holds and there exist functions and such that, for ,Then, for each with , we have(i)if , then (10) has at least one positive periodic solution for each (ii)if , then (10) has at least one positive periodic solution for each , where is some positive constant

By using a fixed point theorem for compact maps on conical shells [31], we established the second positive periodic solution for (1). Recall that a compact operator means an operator which transforms every bounded set into a relatively compact set and introducing the definition of a cone.

Definition 9. Let be a Banach space and let be a closed, nonempty subset of . is a cone if(i) for all and all (ii) implies

Lemma 10 (see [31]). Let be a Banach space and a cone in . Assume are open subsets of with . Letbe a continuous and completely continuous operator such that(i)(ii)There exist such that for and Then, has a fixed point in . The same conclusion remains valid if (i) holds on , and (ii) holds on .

Define

Then, one can readily verify that is a cone in .

Theorem 11. Suppose conditions (H), (H₁)–(H₃) hold. Furthermore, assume that the following two conditions are satisfied:
(H₄) There exist continuous, nonnegative functions , and such thatwhere is nonincreasing and is nondecreasing in
(H₅) There exists such thatThen, problems (1) and (2) have another one positive periodic solution with .

Proof. Let , is given by (25), then, it is easy to verify that is well defined and maps into . Moreover, is continuous and completely continuous, and let be a cone in defined by (59). Define theAs in the proof of Theorem 3.1, we only need to show that (29) has a positive periodic solution with and . We claim that(i)(ii)There exist such that for and We start with (i). In fact, if , then and for all . Fix , thus, we haveTherefore, for each ,...,N. This implies that (i) holds.
Next, we show that (ii) holds. Let , then . Suppose that there exists and such that . Since , then for all . As a result, it follows from (H₄) and (H₅) that, for all ,Hence, ; this is a contradiction and we prove the claim.
Now, Lemma 3.7 guarantees that has at least one fixed point with .

Let us consider again the example (10) in Corollary 7 for the superlinear case.

Corollary 12. Assume in (10) that satisfy (H), for each with . Then, for each with , where is given as in Corollary 7, problem (10) has at least two different positive solutions. To verify (H₄), one may takeand . If , then the existence condition (H₅) becomes

Since β > 1, the right-hand side goes to 0 as . Thus, for any given , it is always possible to find such that (67) is satisfied. Thus, (10) has an additional positive periodic solution .

Remark 13. We emphasize that our results are applicable to the case of a strong singularity as well as the case of a weak singularity since we only need . Moreover, does not need to be positive. In fact, using the assumption that the Green function is positive, one may readily verify that is equivalent to the .
Let us consider the 2-dimensional systemwith

Example 1. Assume that , , , and are positive functions, are given by (69) withThen, the results in Corollary 12 hold.

Proof. We only need show , which is equivalent toSince , a direct computation show that

4. Conclusions

In this paper, we study the periodic problem for nonlinear difference systems with a singularity of repulsive type in the case of . The proofs of main results are based on a nonlinear alternative principle of Leray-Schauder and a fixed point theorem in cones. It is interesting that the singularity is applicable to the case of a weak singularity as well as the case of a strong singularity. In the next research, we will continue to study the periodic problem to the difference systems like (10) where may have attractive singularity at , and whether the condition can be removed.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Hainan Provincial Natural Science Foundation of China (Grant No. 120RC450), National Natural Science Foundation of China (Grant No. 11861028), and Jiangsu Provincial Natural Science Foundation of China (Grant No. BK20201447).

References

Z. Balanov, C. Garca-Azpeitia, and W. Krawcewicz, “On variational and topological methods in nonlinear difference equations,” Communications on Pure & Applied Analysis, vol. 17, no. 6, pp. 2813–2844, 2018.
View at: Publisher Site | Google Scholar
M. A. Delpino and R. F. Manasevich, “Infinitely many T−periodic solutions for a problem arising in nonlinear elasticity,” Journal of Differential Equations, vol. 103, pp. 260–277, 1993.
View at: Publisher Site | Google Scholar
J. Alzabut and C. Tunc, “Existence of periodic solutions for Rayleigh equation with statedependent delay,” Electronic Journal of Differential Equations, vol. 77, pp. 1–8, 2012.
View at: Google Scholar
Z. Cheng and X. Cui, “Positive periodic solution to an indefinite singular equation,” Applied Mathematics Letters, vol. 112, pp. 1–7, 2021.
View at: Google Scholar
A. C. Lazer and S. Solimini, “On periodic solutions of nonlinear differential equations with singularities,” Proceedings of the American Mathematical Society, vol. 99, no. 1, pp. 109–114, 1987.
View at: Publisher Site | Google Scholar
S. Li, F. Liao, and W. Xing, “Periodic solutions of Liénard differential equations with singularity,” Electronic Journal of Differential Equations, vol. 151, pp. 1–12, 2015.
View at: Google Scholar
S. Li, H. Luo, and X. Tang, “Periodic orbits for radially symmetric systems with singularities and semilinear growth,” Results in Mathematics, vol. 72, no. 4, pp. 1991–2011, 2017.
View at: Publisher Site | Google Scholar
I. Rachůnková, M. Tvrdý, and I. Vrkoč, “Existence of nonnegative and nonpositive solutions for second order periodic boundary value problems,” Journal of Differential Equations, vol. 176, pp. 445–469, 2001.
View at: Publisher Site | Google Scholar
F. Wang and Y. Cui, “On the existence of solutions for singular boundary value problem of thirdorder differential equations,” Mathematica Slovaca, vol. 60, no. 4, pp. 485–494, 2010.
View at: Publisher Site | Google Scholar
F. Wang, F. Zhang, and F. Wang, “The existence and multiplicity of positive solutions for second-order periodic boundary value problem,” Journal of Function Spaces and Applications, vol. 2012, article 725646, pp. 1–13, 2012.
View at: Publisher Site | Google Scholar
P. Yan and M. Zhang, “Higher order nonresonance for differential equations with singularities,” Mathematicsl Methods in the Applied Sciences, vol. 26, no. 12, pp. 1067–1074, 2003.
View at: Publisher Site | Google Scholar
Z. Cheng and J. Ren, “Periodic and subharmonic solutions for Duffing equation with a singularity,” Discrete & Continuous Dynamical Systems - A, vol. 32, no. 5, pp. 1557–1574, 2012.
View at: Publisher Site | Google Scholar
D. Franco and J. R. L. Webb, “Collisionless orbits of singular and nonsingular dynamical systems,” Discrete & Continuous Dynamical Systems - A, vol. 15, no. 3, pp. 747–757, 2006.
View at: Publisher Site | Google Scholar
D. Franco and P. J. Torres, “Periodic solutions of singular systems without the strong force condition,” Proceedings of the American Mathematical Society, vol. 136, pp. 1229–1236, 2008.
View at: Publisher Site | Google Scholar
P. J. Torres, “Weak singularities may help periodic solutions to exist,” Journal of Differential Equations, vol. 232, no. 1, pp. 277–284, 2007.
View at: Publisher Site | Google Scholar
M. Zhang, “A relationship between the periodic and the Dirichlet BVPs of singular differential equations,” Proceedings of the Royal Society of Edinburgh, vol. 128, no. 5, pp. 1099–1114, 1998.
View at: Publisher Site | Google Scholar
D. Bonheure and C. De Coster, “Forced singular oscillators and the method of lower and upper solutions,” Topological Methods in Nonlinear Analysis, vol. 22, no. 2, pp. 297–317, 2003.
View at: Publisher Site | Google Scholar
S. Li, F. Liao, and H. Zhu, “Multiplicity of positive solutions to second-order singular differential equations with a parameter,” Boundary Value Problems, vol. 115, 2014.
View at: Publisher Site | Google Scholar
D. Jiang, J. Chu, and M. Zhang, “Multiplicity of positive periodic solutions to superlinear repulsive singular equations,” Journal of Differential Equations, vol. 211, no. 2, pp. 282–302, 2005.
View at: Publisher Site | Google Scholar
J. Alzabut, “Existence of periodic solutions for a type of linear difference equations with distributed delay,” Advances in Difference Equations, vol. 2012, Article ID 53, 2012.
View at: Publisher Site | Google Scholar
F. M. Atici and A. Cabada, “Existence and uniqueness results for discrete second-order periodic boundary value problems,” Computers & Mathematcs with Applications, vol. 45, no. 6-9, pp. 1417–1427, 2003.
View at: Publisher Site | Google Scholar
P. Cheng and X. Tang, “Existence of homoclinic orbits for 2nth-order nonlinear difference equations containing both many advances and retardations,” Journal of Mathematical Analysis and Applications, vol. 381, no. 2, pp. 485–505, 2011.
View at: Publisher Site | Google Scholar
Z. Guo and J. Yu, “The existence of periodic and subharmonic solutions of subquadratic second order difference equations,” Journal of the London Mathematical Society, vol. 68, no. 2, pp. 419–430, 2003.
View at: Publisher Site | Google Scholar
S. Mobayen, “Adaptive global terminal sliding mode control scheme with improved dynamic surface for uncertain nonlinear systems,” International Journal of Control, Automation and Systems, vol. 16, no. 4, pp. 1692–1700, 2018.
View at: Publisher Site | Google Scholar
X. Zhang and X. Tang, “Existence of nontrivial solutions for boundary value problems of second-order discrete systems,” Mathematica Slovaca, vol. 61, no. 5, pp. 769–778, 2011.
View at: Publisher Site | Google Scholar
B. Zhou and C. Liu, “Applications of Conley index theory on difference equations with non-resonance,” Applied Mathematics Letters, vol. 108, article 106500, 2020.
View at: Publisher Site | Google Scholar
J. Chu, P. J. Torres, and M. Zhang, “Periodic solutions of second order non-autonomous singular dynamical systems,” Journal of Differential Equations, vol. 239, no. 1, pp. 196–212, 2007.
View at: Publisher Site | Google Scholar
S. Li and Y. Wang, “Multiplicity of positive periodic solutions to second order singular dynamical systems,” Mediterranean Journal of Mathematics, vol. 14, article 202, 2017.
View at: Publisher Site | Google Scholar
F. M. Atici and G. S. Guseinov, “Positive periodic solutions for nonlinear difference equations with periodic coefficients,” Journal of Mathematical Analysis and Applications, vol. 232, no. 1, pp. 166–182, 1999.
View at: Publisher Site | Google Scholar
D. O’Regan, Existence Theory for Nonlinear Ordinary Differential Equations, Kluwer Academic, Dordrecht, 1997.
View at: Publisher Site
D. Guo and V. Lakshmikantham, Nonlinear Problem in Abstract Cones, Academic Press, London, 1988.

Copyright

Copyright © 2021 Shengjun Li and Fang Zhang. 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.

PDF Download Citation

Download other formats

Order printed copies

Views

148

Downloads

442

Citations