Abstract
In this article, we consider two nonlinear neutral systems with multiple delays. Our main tool here is to use dichotomy theory to construct an implicit solution for these two systems. Utilizing Krasnoselskii’s fixed point theorem, we obtain sufficient criteria for the existence of periodic solutions, as well as for the uniqueness of solutions. The main results expand and generalize certain previously published findings.
1. Introduction
Periodic solutions of equations are solutions that describe regularly repeated processes. The periodic solutions of systems of differential equations occupy special importance in branches of science such as the theory of oscillations, dynamical systems, and celestial mechanics, and the analysis of these systems in depth opens up new possibilities and horizons in these sciences. Such a study aids in understanding the geometric behavior of solutions eventually (see [1–4]).
In recent years, several investigators have tried the stability and existence of periodic solutions by using the technique of fixed point, in particular Burton, Furumochi, Zhang, and others (see [5–13]).
By Krasnoselskii’s fixed point theorem, Luo et al. [14] investigate the existence of positive periodic solutions for two neutral functional differential equations in which ; ; ; ; , , , and are -periodic functions; and are constants; ; and .
The above functional differential equations ( (1) and (2)) cover many mathematical ecological and population models, for example, hematopoiesis models (see [15, 16]), Nicholson’s blowflies models (see [17, 18]), and blood cell production (see [19]).
Sa Ngiamsunthorn [11] considered the differential system with dichotomy condition (3) periodic coefficients. Similar system of (3) has been studied in [20].
Motivated by the works mentioned above, we are concerned with the existence of periodic solutions for two nonlinear neutral systems of differential equations in which , , and , , are real continuous -periodic functions on , . is a real continuous matrix -periodic function defined on . is a real continuous matrix periodic function defined on with . The functions and are real continuous vector functions defined on and , respectively, such that
Note that the functional and function are in different spaces because is in the phase space, but their norms are equivalent (for more details on space theory, we refer the reader to the following papers) [21, 22].
This paper is arranged as follows: after this introduction, we list a set of definitions and previous results related to integrable dichotomies and fixed point theorems in Section 2. Sections 3 and 4 deal with the existence and uniqueness of periodic solutions of systems (4) and (5), respectively, and are followed by a conclusion.
2. Preliminaries
In this section, we outline some results and definitions of integrable dichotomy that will be crucial in the proof of our results (see [23, 24]). Consider the following linear differential system:
in which is a continuous matrix function. Let be the fundamental matrix solution of system (7) with . Assume is a projection matrix. We let a green matrix be associated with by
Definition 1 (see [23]). If a projection matrix and a positive constant exist for which the associated Green matrix satisfies the linear differential system (7) has an integrable dichotomy.
Proposition 2 (see [23]). Assume that system (7) has an integrable dichotomy. Then, is the only bounded solution to (7).
Now, the set of bounded and continuous functions is designated as . If we consider the nonhomogeneous linear system
under an integrable dichotomy condition, we take the following theorem from [23].
Theorem 3. Assume that system (7) has an integrable dichotomy. If , then system (10) has a unique bounded solution . Furthermore,
Theorem 4 (see [23]). Assume that the homogeneous system (7) has an integrable dichotomy for which is bounded. If is -periodic, then is also -periodic. In addition, if is -periodic, then (10) has a unique periodic solution satisfying (11).
We present the fixed point theorems that we utilize to demonstrate the existence and uniqueness of periodic solutions to system (4) (see [5, 25]).
Theorem 5 (Banach). Assume that is a complete metric space and . If there is a constant such that for , then there is one and only one point with .
Smart [25] established a hybrid result by combining Banach’s theorem and Schauder’s theorem as follows:
Theorem 6 (Krasnoselskii). Let be a closed bounded convex nonempty subset of a Banach space . Assume that and map into such that (i) is a contraction mapping on (ii) is completely continuous on (iii) implies Then, there exists with .
Assume be a constant. Denote
Clearly, the set is a bounded nonempty closed and convex subset of .
Assume that, for , there exists such that and for , there exists such that
Denote , , and , and we assume also
3. Existence of Periodic Solutions for (4)
In this section, we show the existence and the uniqueness of solution of (4) under the conditions stated in the previous section. So, let
Hence,
By Theorem 3, system (4) holds the integral equation The above equation is equivalent to
Define the operators and by
Note that if the operator has a fixed point, then this fixed point is a periodic solution of (4).
Lemma 7. If (14) and (15) hold, then the operators and are defined by (21) and (22), respectively, from into , that is, .
Proof. Let , by (14). Therefore, Secondly, for , by (14) and (15), we get Since all quantities in and are periodic, then .
Lemma 8. If (14) holds, then the operator defined by (21) is a contraction.
Proof. Let . By using (14), we get Then, Therefore, is a contraction because .
Lemma 9. If (14) and (15) hold, then the operator defined by (22) is completely continuous.
Proof. To prove the operator completely continuous, we must prove that is continuous and is contained in a compact set; for this purpose, let where is a positive integer such that as . Then, So, the dominated convergence theorem implies which implies that is continuous. Next, we show that the image of is contained in a compact set. Let , and by (24), we have Second, we calculate and show that it is uniformly bounded. Then, Thus, the sequence is uniformly bounded and equicontinuous. As a result, by Ascoli-Arzela’s theorem is relatively compact.
We next prove for any that .
Lemma 10. If (14)–(16) hold, then for any , we have .
Proof. Let . Then, . By (16), we have It follows that for all . Hence, .
Theorem 11. Assume that system (7) has an integrable dichotomy. If conditions (14)–(16) hold, then system (4) has at least one -periodic solution.
Proof. Clearly, by Lemmas 7–10, all the requirements of the Krasnoselskii’s theorem are satisfied. Thus, there exists a fixed point such that ; this fixed point is a solution of (4). Hence, (4) has a -periodic solution.
Theorem 12. Assume that system (7) has an integrable dichotomy. If conditions (14) and (15) and hold, then system (4) has a unique -periodic solution.
Proof. Let the mapping be presented by For , we obtain Since (34) hold, the contraction mapping completes the proof.
Example 1. Consider system (4) with , , , and , and
Let the set
Clearly, the set is a bounded nonempty closed and convex subset of for any positive constant .
Note that , , , and , and we use to get
Then, .
We can see that conditions (14) and (15) hold.
We substitute all quantities in the inequality (16), and we have
Now, since the matrix is continuous and periodic, then system (4) has an integrable dichotomy, and we have two cases: if , then (16) holds for any positive constant , and by Theorem 11, system (4) has at least one -periodic solution.
If , then condition (34) holds, and by Theorem 11, system (4) has a unique -periodic solution.
4. Existence of Periodic Solutions for (5)
In this section, we show the existence and the uniqueness of the solution of (5) under the conditions stated in the previous section. So, let Then,
By Theorem 3, system (5) holds the integral equation The above equation is equivalent to
We define, for the operators and by
Lemma 13. If (14) and (15) hold, then the operators and defined above are operators from into , that is, .
Proof. Let ; by (14), we get
Secondly, for , by (14) and (15), we get
Since all quantities in and are periodic, then .
By the same technique proofs in Lemmas 8–10, we state the following lemmas without proofs.
Lemma 14. If (14) holds, then the operator defined by (45) is a contraction.
Lemma 15. If (14) and (15) hold, then the operator defined by (46) is completely continuous.
Lemma 16. If (14)–(16) hold, then for any , we have .
Theorem 17. Assume that system (7) has an integrable dichotomy. If conditions (14)–(16) hold, then system (5) has at least one -periodic solution.
Proof. By Lemmas 13–16, all the requirements of the Krasnoselskii’s theorem are satisfied. Thus, there exists a fixed point such that ; this fixed point is a solution of (5). Hence, (5) has a -periodic solution.
Theorem 18. Assume that system (7) has an integrable dichotomy. If conditions (14), (15), and (34) hold, then system (5) has a unique -periodic solution.
Proof. Let the mapping be presented by For , we have Since (34) holds, the contraction mapping completes the proof.
5. Conclusion
In this paper, we dealt with the study of types of neutral equations more generally, represented in nonlinear systems with several delays under the dichotomy condition, where the fixed point theorems were used to prove existence and uniqueness.
The benefit of this paper is to generalize several well-known researches such as [11, 14]. So that if , then our results will apply to systems (3) of [11]. Also, the periodicity of (1) and (2) in [14] is generalized by our systems (4) and (5) in -dimentional case.
Data Availability
The numerical data used to support the findings of this study are included in the article.
Conflicts of Interest
The authors declare that they have no competing interests. There are no any nonfinancial competing interests (political, personal, religious, ideological, academic, intellectual, commercial, or any other) to declare in relation to this manuscript.
Authors’ Contributions
Mesmouli directed the study and helped inspection. All the authors carried out the main results of this article and drafted the manuscript and read and approved the final manuscript.
Acknowledgments
This research has been funded by the Scientific Research Deanship at University of Hail, Saudi Arabia, through project number RG-20 125.