Dichotomy Condition and Periodic Solutions for Two Nonlinear Neutral Systems

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 ﬁ xed point theorem, we obtain su ﬃ cient criteria for the existence of periodic solutions, as well as for the uniqueness of solutions. The main results expand and generalize certain previously published ﬁ ndings.


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][2][3][4]).
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 y : ℝ ⟶ ℝ n , τðζÞ, and σ i ðζÞ, i = 1, ⋯, m, are real continuous T-periodic functions on ℝ, T > 0. AðζÞ is a n × n real continuous matrix T-periodic function defined on ℝ . QðωÞ is a n × n real continuous matrix periodic function defined on ð−∞, 0 with Ð 0 −∞ QðωÞdω = I. The functions qð ζ, uÞ and f ðζ, u 1 , ⋯, u m Þ are real continuous vector functions defined on ℝ × ℝ n and ℝ × ðℝ n Þ m , respectively, such that Note that the functional yðζ − τðζÞÞ and function yðζÞ are in different spaces because yðζ − τðζÞÞ 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.

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 AðζÞ is a continuous n × n matrix function. Let ΨðζÞ be the fundamental matrix solution of system (7) with Ψð0Þ = I. Assume P is a projection matrix. We let a green matrix G ≔ G P be associated with P by Definition 1 (see [23]). If a projection matrix P and a positive constant μ exist for which the associated Green matrix G = G P satisfies the linear differential system (7) has an integrable dichotomy.
Proposition 2 (see [23]). Assume that system (7) has an integrable dichotomy. Then, zðζÞ = 0 is the only bounded solution to (7). Now, the set of bounded and continuous functions is designated as BCðℝ, ℝ n Þ. If we consider the nonhomogeneous linear system under an integrable dichotomy condition, we take the following theorem from [23].
We present the fixed point theorems that we utilize to demonstrate the existence and uniqueness of periodic solutions to system (4) (see [5,25]).
then there is one and only one point z ∈ Y with Γz = z.
Smart [25] established a hybrid result by combining Banach's theorem and Schauder's theorem as follows: Journal of Function Spaces Theorem 6 (Krasnoselskii). Let Ω be a closed bounded convex nonempty subset of a Banach space Y. Assume that Γ 1 and Clearly, the set Ω is a bounded nonempty closed and convex subset of BCðℝ, ℝ n Þ.
Proof. To prove the operator Γ 2 : Ω ⟶ BCðℝ, ℝ n Þ completely continuous, we must prove that Γ 2 is continuous and Γ 2 ðΩÞ is contained in a compact set; for this purpose, let u n ∈ Ω where n is a positive integer such that u n ⟶ u as n ⟶ ∞. Then, So, the dominated convergence theorem implies which implies that Γ 2 is continuous. Next, we show that the image of Γ 2 is contained in a compact set. Let u n ∈ Ω, and by (24), we have Second, we calculate ðΓ 2 u n Þ ′ ðζÞ and show that it is uniformly bounded. Then, Thus, the sequence ðΓ 2 u n Þ is uniformly bounded and equicontinuous. As a result, by Ascoli-Arzela's theorem Γ 2 ðΩÞ is relatively compact.
Proof. Clearly, by Lemmas 7-10, all the requirements of the Krasnoselskii's theorem are satisfied. Thus, there exists a fixed point z ∈ Ω such that z = Γ 1 z + Γ 2 z; this fixed point is a solution of (4). Hence, (4) has a T-periodic solution.
We can see that conditions (14) and (15) hold. We substitute all quantities in the inequality (16), and we have Now, since the matrix A is continuous and periodic, then system (4) has an integrable dichotomy, and we have two cases: if μ ≤ ðð1 − 10 −4 Þ/ð3 × 10 −5 ÞÞ, then (16) holds for any positive constant M, and by Theorem 11, system (4) has at least one 2π-periodic solution.