Positive Periodic Solutions for a Class of Strongly Coupled Differential Systems with Singular Nonlinearities

Chen, Ruipeng; Zhang, Guangchen; Liu, Jiayin

doi:https://doi.org/10.1155/2021/5964540

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

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Reinforcement Learning and Adaptive Optimisation of Complex Dynamic Systems and Industrial Applications

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Research Article | Open Access

Volume 2021 | Article ID 5964540 | https://doi.org/10.1155/2021/5964540

Positive Periodic Solutions for a Class of Strongly Coupled Differential Systems with Singular Nonlinearities

Ruipeng Chen,¹Guangchen Zhang,¹and Jiayin Liu¹

Academic Editor: Dan Selisteanu

Received13 Jun 2020

Revised01 Oct 2020

Accepted28 Dec 2020

Published16 Jan 2021

Abstract

This article studies the existence of positive periodic solutions for a class of strongly coupled differential systems. By applying the fixed point theory, several existence results are established. Our main findings generalize and complement those in the literature studies.

1. Introduction

In this paper, we are concerned with the existence of positive periodic solutions of the strongly coupled differential systems:where is a linear differential operator with . In addition, we assume and , that is, is a -Caratheodory function, and it is singular at .

During the past few decades, the fixed point theory has been widely adopted to investigate the nonperiodic coupled differential systems, and researchers have mainly concentrated on the existence and multiplicity of positive solutions [1–3]. Meanwhile, the periodic equations and systems with singular nonlinearities have been dealt via some classical fixed point theorems, such as Schauder’s fixed point theorem and fixed point theorems in cones [4–12]. What is worth mentioning is the results obtained in [5, 6, 11, 12], where the authors show, under some circumstances, weak singularities are helpful to seek out periodic solutions for not only singular equations [5] but also singular coupled systems [11]. Especially, in [10], Li and Zhang considered the singular equationwhere and . By employing a fixed point theorem in cones, they established several existence theorems under the following basic assumption.

() There exist , and such thatand pointed out they have not limited themselves to the weak singularities; see [10], Section 3, for more details. Besides, the case (the resonant case) has also been studied in [10], Theorem 4.1. For other research works related to the resonant case of (2), one may refer to [13–15] and references therein.

To our knowledge, however, so far, the existing results on strongly coupled periodic singular systems are relatively few. Therefore, motivated by the aforementioned papers, we shall establish the existence of positive periodic solutions of system (1) in the present paper to further improve and complement those in the literature studies. To demonstrate our new results, we choose the following differential systems:where and . Here, means we need not restrict ourselves to the weak force conditions in our results.

The rest of the paper is arranged as follows. In Section 2, we give some required preliminaries and notations. In Section 3, we shall state and prove the existence results for (1) in the nonresonant case. Finally, in Section 4, an existence theorem will be proved for (1) in the resonant case .

2. Preliminaries

The linear boundary value problemis called nonresonant if its unique solution is the trivial one. When (5) and (6) are nonresonant, the well-known Fredholm’s alternative ensures the nonhomogeneous equationadmits a unique -periodic solution, which can be expressed aswhere is Green’s function associated to (5) and (6).

For given , we denote by and , respectively, the essential infimum and supremum of . means for , and it is positive on a set with positive measures. Moreover, if (7) has a unique periodic solution for any and is positive on when , then we say (5) satisfies the antimaximum principle. Recently, Hakl and Torres [16] established an explicit criterion to guarantee the antimaximum principle holds for (5). For the sake of convenience, set

Lemma 1 (see [16]). If andthen (5) satisfies the antimaximum principle, where , whereafter Chu et al. [17] pointed out if (5) admits the antimaximum principle, then on . In addition, they obtained the following.

Lemma 2. If and (11) holds, then the distance between two consecutive zeroes of a nontrivial solution of (5) is greater than .

Obviously, Lemma 2 implies does not vanish. As a consequence of Lemmas 1 and 2, Chu et al. established the following.

Lemma 3. If and (10) and (11) hold, then on .

Note that Lemma 3 plays an important role in the application of the classical fixed point theorems. Indeed, by Lemma 3, the positivity of some completely continuous operators could be easily obtained.

Remark 1. Clearly, if (without damping terms), then (10) and (11) reduce, respectively, towhich are conditions used to guarantee the positivity of Green’s function corresponds to (5) and (6); see [18] for more details.
Throughout the paper, we always suppose and . Furthermore, (H1) The linear equation is nonresonant, and corresponding Green’s function on . (H2) There are , and so that, for ,It is not hard to see (H1) implies the antimaximum principle holds for , and thus, , whereHence,Set , where and ). For , we write . Letbe the Banach space equipped with the normHere, .
A vector function is called a positive -periodic solution of (1) if it satisfies (1) and on , . LetThen, it is not difficult to check is a positive cone in , and for any , we get by (15) thatFor , let ; then, .

Lemma 4 (see [19]). Let be a Banach space and a cone. Suppose are open bounded subsets satisfying . If is completely continuous and satisfies there is such that for and , ,or there is such that for and , ,then admits a fixed point in .

3. The Nonresonant Systems

This section is devoted to establishing the existence results for system (1) in the nonresonant case. To this end, we define

Theorem 1. Assume (H1) and (H2) hold. Let(i)If , , and then (1) has a positive -periodic solution.(ii)If , then (1) has a positive -periodic solution.

Proof. (i)Let be an operator defined by with Then, is completely continuous, and a -periodic solution of (1) is equivalent to a fixed point of . We shall divide the proof of the case into three steps as follows. Step 1: if we choose , then for , Or else, if there are and so that , then by the definition of and (15), we get This together with (H2) implies which contradicts since . Step 2: for , maps into . Since , for any satisfying ] and , we can deduce from (H2) that and subsequently, for , which means maps into . Step 3: we shall show Recall that . For any , we have , and then by (27), (H1), and (H2), we can obtain which yields . This together with (22) shows And accordingly, Lemma 4 ensures (1) admits a positive -periodic solution.(ii)To deal with this case, we only need to show (24)–(29) are still satisfied. Indeed, if we choose , then we can easily prove by (H2) that (24) holds true. Moreover, since , could be chosen so that (27) and (29) are all satisfied. Consequently, Lemma 4 yields (1) has a positive -periodic solution.

Remark 2. Obviously, (H2) reduces to () under some special circumstances; hence, Theorem 1 generalizes [10], Theorem 3.1.

Remark 3. When , , and , system (1) becomesIn [11], several existence theorems have been established for (32) via Schauder’s fixed point theorem, where satisfies only the weak force conditions. However, we do not restrict ourselves here to weak singularities, and Theorem 1 is still valid for (32) with strong singularities.

Theorem 2. Assume (H1) and (H3) There are , and such that, for ,

Then, the following results hold true:(i)If , , and then (1) has a positive -periodic solution.(ii)If , then (1) has a positive -periodic solution.

Here, is defined as in Theorem 1.

Proof. (i)Choose and set . By (H3) and an argument similar to the proof of Theorem 1 (i), we can get (24) and (27). To apply Lemma 4, we just need to show In fact, for any , it follows from (27) and (H3) that and then (34) yields . Therefore, Lemma 4 implies (1) has a positive -periodic solution.(ii)Using Lemma 4 and similar to the proof of Theorem 1 (ii), we can easily get the conclusions.

Remark 4. Jiang et al. [4] studied the singular equationwhere are constants, are nonnegative, and . For the positone case, they supposed , which means , and for the semipositone case, a strong force condition was required. See [4], Corollaries 3.2 and 4.3. Evidently, Theorem 2 generalizes [10], Theorem 3.2, and the corresponding ones in [4].

4. The Resonant Systems

Let us consider the singular systems

Theorem 3. Assume (H2) and (H4) There are so that, for any and ,If and , then (38) has a positive -periodic solution.

Proof.
Let us take into account the auxiliary systemswhere are constants introduced as in (H4). Clearly, a solution of (40) is just a solution of original system (38), and vice versa. Therefore, to complete the proof, it is enough to show (40) has a positive -periodic solution.
Let be Green’s function of Hill’s equation . Then, a solution of (40) is equivalent to a fixed point of completely continuous operator with components :Moreover, let and denote the minimum and maximum of , respectively; then, , and so, . For , we denote again by , introduced in (18), the positive cone in .
Choose sufficiently large and define . Since , we have for ,and subsequently, by (H4),this yields .
By (H2) and , there exists small enough such that, for any satisfying ,which impliesLet and ; then,Otherwise, if there are and such that , then from (45), it followsSetting , we obtain from (47) that , which contradicts .
To apply Lemma 4, it remains to verifyUsing (H4), we can easily getand thus,which means (48) holds true.
Consequently, Lemma 4 ensures (40) has a positive -periodic solution, and accordingly, system (38) admits a positive -periodic solution.

Remark 5. Many authors have paid their attention to the optimal control of the nonlinear systems, and a number of excellent results have been established. See, for instance, [20–23] and the references therein. For the optimal control of nonlinear system (1), we shall deal in the forthcoming paper.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The corresponding author was supported by the National Natural Science Foundation of China (no. 61761002) and the NSF of Ningxia Hui Autonomous Region of China (nos. 2020AAC03232 and 2020AAC03234).

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Copyright © 2021 Ruipeng Chen 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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