# Dichotomy Condition and Periodic Solutions for Two Nonlinear Neutral Systems

Authors: Mouataz Billah Mesmouli; Adel A. Attiya; A. A. Elmandouha; Ayékotan M. J. Tchalla; Taher S. Hassan
Journal: Journal of Function Spaces
 (2022)

---

## 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.


---

## Body

## 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
(1)yζ−cyζ−τζ′=−aζyζ+fζ,yζ−τζ,(2)ddζyζ−c∫−∞0Qωyζ+ωdω=−aζyζ+∫−∞0Qωfζ,yζ+ωdω,in which y:ℝ⟶ℝ; aζ∈Cℝ,0,∞; f∈Cℝ×ℝ,ℝ; τζ∈Cℝ,ℝ; aζ, bζ, τζ, and fζ,y are T-periodic functions; T>0 and c<1 are constants; Qω∈C−∞,0,0,∞; and ∫−∞0Qωdω=1.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
(3)yζ−cyζ−τ′=Aζyζ+fζ,yζ−σ1ζ,⋯,yζ−σmζ,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(4)yζ−qζ,yζ−τζ′=Aζyζ+fζ,yζ−σ1ζ,⋯,yζ−σmζ,(5)yζ−∫−∞0Qωqζ,yζ+τωdω′=Aζyζ+∫−∞0Qωfζ,yζ+σ1ω,⋯,yζ+σmωdω,in which y:ℝ⟶ℝn, τζ, and σiζ, i=1,⋯,m, are real continuous T-periodic functions on ℝ, T>0. Aζ is a n×n real continuous matrix T-periodic function defined on ℝ. Qω is a n×n real continuous matrix periodic function defined on −∞,0 with ∫−∞0Qωdω=I. The functions qζ,u and fζ,u1,⋯,um are real continuous vector functions defined on ℝ×ℝn and ℝ×ℝnm, respectively, such that
(6)fζ+T,u1,u2,⋯,um=fζ,u1,u2,⋯,um,qζ+T,u=qζ,u.Note that the functionalyζ−τζ and function yζ are in different spaces because yζ−τζ 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 Section2. 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:
(7)z′ζ=Aζzζ,in whichAζ is a continuous n×n matrix function. Let Ψζ be the fundamental matrix solution of system (7) with Ψ0=I. Assume P is a projection matrix. We let a green matrix G≔GP be associated with P by
(8)Gζ,ω=ΨζPΨ−1ζ,forζ≥ω,−ΨζI−PΨ−1ζ,forζ<ω.Definition 1 (see [23]).
If a projection matrixP and a positive constant μ exist for which the associated Green matrix G=GP satisfies
(9)supζ∈ℝ∫−∞∞Gζ,ωdω=μ,the linear differential system (7) has an integrable dichotomy.Proposition 2 (see [23]).
Assume that system (7) has an integrable dichotomy. Then, zζ=0 is the only bounded solution to (7).
Now, the set of bounded and continuous functions is designated asBCℝ,ℝn. If we consider the nonhomogeneous linear system
(10)z′ζ=Aζzζ+fζ,under an integrable dichotomy condition, we take the following theorem from [23].Theorem 3.
Assume that system (7) has an integrable dichotomy. If f∈BCℝ,ℝn, then system (10) has a unique bounded solution z∈BCℝ,ℝn. Furthermore,
(11)zζ=∫−∞∞Gζ,ωfωdω.Theorem 4 (see [23]).
Assume that the homogeneous system (7) has an integrable dichotomy for which ΨζPΨ−1ζ is bounded. If A is T-periodic, then ΨζPΨ−1ζ is also T-periodic. In addition, if f∈BCℝ,ℝn is T-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 thatY,ρ is a complete metric space and Γ:Y⟶Y. If there is a constant γ<1 such that for u,v∈Y,
(12)ρΓu,Γv≤γρu,v,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:Theorem 6 (Krasnoselskii).
LetΩ be a closed bounded convex nonempty subset of a Banach space Y. Assume that Γ1 and Γ2 map Ω into Y such that
(i)
Γ1 is a contraction mapping on Ω(ii)
Γ2 is completely continuous on Ω(iii)
u,v∈Ω implies Γ1u+Γ2v∈Ω
Then, there existsz∈Ω with z=Γ1z+Γ2z.AssumeM>0 be a constant. Denote
(13)Ω=u∈BCℝ,ℝn:u≤Manduζ+T=uζforallζ∈ℝ.Clearly, the setΩ is a bounded nonempty closed and convex subset of BCℝ,ℝn.Assume that, foru,v∈Ω, there exists L1∈0,1 such that
(14)qζ,u−qζ,v≤L1u−v,forallζ∈ℝ,and for u1,u2,⋯um,v1,v2,⋯,vm∈Ω, there exists L2>0 such that
(15)fζ,u1,u2,⋯,um−fζ,v1,v2,⋯,vm≤L2u1−v1+⋯+um−vm,forallζ∈ℝ.Denotesupζ∈0,Tqζ,0=α, supζ∈0,Tfζ,0,⋯,0=β, and supζ∈0,TAζ=λ, and we assume also
(16)L1M+α+μλL1M+α+L2mM+β≤M.


## 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
(17)zζ=yζ−qζ,yζ−τζ.Hence,(18)z′ζ=Aζzζ+Aζqζ,yζ−τζ+fζ,yζ−σ1ζ,⋯,yζ−σmζ.By Theorem3, system (4) holds the integral equation
(19)zζ=∫−∞∞Gζ,ωAωqω,yω−τω+fω,yω−σ1ω,⋯,yω−σmωdω.The above equation is equivalent to
(20)yζ=qζ,yζ−τζ+∫−∞∞Gζ,ωAωqω,yω−τζ+fω,yω−σ1ω,⋯,yω−σmωdω.Define the operatorsΓ1 and Γ2 by
(21)Γ1uζ≔qζ,uζ−τζforu∈BCℝ,ℝn,(22)Γ2uζ≔∫−∞∞Gζ,ωAωqω,uω−τωdω+∫−∞∞Gζ,ωfω,uω−σ1ω,⋯,uω−σmωdω.Note that if the operatorΓ1+Γ2 has a fixed point, then this fixed point is a periodic solution of (4).Lemma 7.
If (14) and (15) hold, then the operators Γ1 and Γ2 are defined by (21) and (22), respectively, from Ω into BCℝ,ℝn, that is, Γ1,Γ2:Ω⟶BCℝ,ℝn.Proof.
Letu∈Ω, by (14). Therefore,
(23)Γ1uζ=qζ,uζ−τζ≤L1uζ−τζ+qζ,0≤L1u+supζ∈0,ζqζ,0≤L1M+α.
Secondly, foru∈Ω, by (14) and (15), we get
(24)Γ2uζ=∫−∞∞Gζ,ωAωqω,uω−τωdω+∫−∞∞Gζ,ωfω,uω−σ1ω,⋯,uω−σmωdω≤λL1M+α+L2mM+β∫−∞∞Gζ,ωdω≤μλL1M+α+L2mM+β.
Since all quantities inΓ1 and Γ2 are periodic, then Γ1,Γ2:Ω⟶BCℝ,ℝn.Lemma 8.
If (14) holds, then the operator Γ1:Ω⟶BCℝ,ℝn defined by (21) is a contraction.Proof.
Letu,v∈Ω. By using (14), we get
(25)Γ1uζ−Γ1vζ=qζ,uζ−τζ−qζ,vζ−τζ≤L1uζ−τζ−vζ−τζ≤L1u−v.
Then,(26)Γ1u−Γ1v≤L1u−v.
Therefore,Γ1 is a contraction because L1∈0,1.Lemma 9.
If (14) and (15) hold, then the operator Γ2:Ω⟶BCℝ,ℝn defined by (22) is completely continuous.Proof.
To prove the operatorΓ2:Ω⟶BCℝ,ℝn completely continuous, we must prove that Γ2 is continuous and Γ2Ω is contained in a compact set; for this purpose, let un∈Ω where n is a positive integer such that un⟶u as n⟶∞. Then,
(27)Γ2unζ−Γ2uζ≤∫−∞∞Gζ,ωAωqω,unω−τω−qω,uω−τωdω+∫−∞∞Gζ,ω×fω,unω−σ1ω,⋯,unω−σmω−fω,uω−σ1ω,⋯,uω−σmωdω≤μλL1+L2mun−u.
So, the dominated convergence theorem implies(28)limn⟶∞Γ2unζ−Γ2uζ=0,which implies that Γ2 is continuous. Next, we show that the image of Γ2 is contained in a compact set. Let un∈Ω, and by (24), we have
(29)Γ2un≤μλL1M+α+L2mM+β.
Second, we calculateΓ2un′ζ and show that it is uniformly bounded.
(30)Γ2un′ζ=∫−∞∞Gζ,ωAωqω,unω−τω+fω,unω−σ1ω,⋯,unω−σmωdω′=un′ζ−q′ζ,unζ−τζ=Aζuζ+fζ,unζ−σ1ζ,⋯,unζ−σmζ=AζΓ1un+Γ2un+fζ,unζ−σ1ζ,⋯,unζ−σmζ.
Then,(31)Γ2un′≤λ+L2mM+β.
Thus, the sequenceΓ2un is uniformly bounded and equicontinuous. As a result, by Ascoli-Arzela’s theorem Γ2Ω is relatively compact.We next prove for anyu,v∈Ω that Γ1u+Γ2v∈Ω.Lemma 10.
If (14)–(16) hold, then for any u,v∈Ω, we have Γ1u+Γ2v∈Ω.Proof.
Letu,v∈Ω. Then, u,v≤M. By (16), we have
(32)Γ1uζ+Γ2vζ≤qζ,uζ−τζ+∫−∞∞Gζ,ωAωqω,vω−τωdω+∫−∞∞Gζ,ωfω,vω−σ1ω,⋯,vω−σmωdω≤L1M+α+λL1M+α+L2mM+β∫−∞∞Gζ,ωdω≤L1M+α+μλL1M+α+L2mM+β≤M.
It follows that(33)Γ1u+Γ2v≤M,for all u,v∈Ω. Hence, Γ1u+Γ2v∈Ω.Theorem 11.
Assume that system (7) has an integrable dichotomy. If conditions (14)–(16) hold, then system (4) has at least one T-periodic solution.Proof.
Clearly, by Lemmas7–10, all the requirements of the Krasnoselskii’s theorem are satisfied. Thus, there exists a fixed point z∈Ω such that z=Γ1z+Γ2z; this fixed point is a solution of (4). Hence, (4) has a T-periodic solution.Theorem 12.
Assume that system (7) has an integrable dichotomy. If conditions (14) and (15) and
(34)L1+μλL1+L2m<1,hold, then system (4) has a unique T-periodic solution.Proof.
Let the mappingΓ be presented by
(35)Γuζ=qζ,uζ−τζ+∫−∞∞Gζ,ωAωqω,uω−τω+fω,uω−σ1ω,⋯,uω−σmωdω.
Foru1,u2∈BCℝ,ℝn, we obtain
(36)Γu1ζ−Γu2ζ≤qζ,u1ζ−τζ−qζ,u2ζ−τζ+∫−∞∞Gζ,ωAωqω,u1ω−τω−qω,u2ω−τωdω+∫−∞∞Gζ,ω×fω,u1ω−σ1ω,⋯,u1ω−σmω−fω,u2ω−σ1ω,⋯,u2ω−σmωdω=L1+μλL1+L2mu1−u2.
Since (34) hold, the contraction mapping completes the proof.Example 1.
Consider system (4) with n=2, m=2, T=2π, and y=y1,y2t, and
(37)yζ=y1ζy2ζ,qζ,yζ−τζ=10−4sinζy2ζ−cosζy1ζ−cosζ,Aζ=10−2sint−0.990.9910−3sint,fζ,yζ−σ1ζ,yζ−σ2ζ=10−5cosζy1ζ−10−2+y2ζ−sinζy2ζ−10−2+y1ζ−sinζ.
Let the set(38)Ω=u∈BC0,2π,ℝn:u≤Manduζ+2π=uζforallζ∈ℝ.
Clearly, the setΩ is a bounded nonempty closed and convex subset of BC0,2π,ℝn for any positive constant M.
Note thatL1=10−4, L2=10−5, α=0, and β=0, and we use A=max1≤j≤2∑i=12aij to get
(39)Aζ=10−2sinζ−0.990.9910−3sinζ=max10−2sinζ+0.99,10−3sinζ+0.99=10−2sinζ+0.99.
Then,λ=10−1.
We can see that conditions (14) and (15) hold.
We substitute all quantities in the inequality (16), and we have
(40)10−4+μ10−5+2×10−5≤1.Now, since the matrixA is continuous and periodic, then system (4) has an integrable dichotomy, and we have two cases: if μ≤1−10−4/3×10−5, then (16) holds for any positive constant M, and by Theorem 11, system (4) has at least one 2π-periodic solution.Ifμ<1−10−4/3×10−5, then condition (34) holds, and by Theorem 11, system (4) has a unique 2π-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
(41)zζ=yζ−∫−∞0Qrqζ,yζ+τrdr.Then,
(42)z′ζ=Aζzζ+Aζ∫−∞0Qrqζ,yζ+τrdr+∫−∞0Qrfζ,yζ+σ1r,⋯,yζ+σmrdr.By Theorem3, system (5) holds the integral equation
(43)zζ=∫−∞∞Gζ,ωAω∫−∞0Qrqω,yω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,yω+σ1r,⋯,yω+σmrdrdω.The above equation is equivalent to
(44)yζ=∫−∞0Qrqζ,yζ+τrdr+∫−∞∞Gζ,ωAω∫−∞0Qrqω,yω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,yω+σ1r,⋯,yω+σmrdrdω.We define, foru∈BCℝ,ℝn, the operators Y1 and Y2 by
(45)Y1uζ=∫−∞0Qrqζ,uζ+τrdr,(46)Y2uζ=∫−∞∞Gζ,ωAω∫−∞0Qrqω,uω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,uω+σ1r,⋯,uω+σmrdrdω.Lemma 13.
If (14) and (15) hold, then the operators Y1 and Y2 defined above are operators from Ω into BCℝ,ℝn, that is, Y1,Y2:Ω⟶BCℝ,ℝn.Proof.
Letu∈Ω; by (14), we get
(47)Y1uζ=∫−∞0Qrqζ,uζ+τrdr≤L1uζ−τζ+qζ,0∫−∞0Qrdr≤L1u+supζ∈0,ζqζ,0I≤L1M+α.
Secondly, foru∈Ω, by (14) and (15), we get
(48)Y2uζ=∫−∞∞Gζ,ωAω∫−∞0Qrqω,uω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,uω+σ1r,⋯,uω+σmrdrdω≤λL1M+α+L2mM+β∫−∞∞Gζ,ω∫−∞0Qrdrdω≤μλL1M+α+L2mM+β.
Since all quantities inY1 and Y2 are periodic, then Y1,Y2:Ω⟶BCℝ,ℝn.
By the same technique proofs in Lemmas8–10, we state the following lemmas without proofs.Lemma 14.
If (14) holds, then the operator Y1:Ω⟶BCℝ,ℝn defined by (45) is a contraction.Lemma 15.
If (14) and (15) hold, then the operator Y2:Ω⟶BCℝ,ℝn defined by (46) is completely continuous.Lemma 16.
If (14)–(16) hold, then for any u,v∈Ω, we have Y1u+Y2v∈Ω.Theorem 17.
Assume that system (7) has an integrable dichotomy. If conditions (14)–(16) hold, then system (5) has at least one T-periodic solution.Proof.
By Lemmas13–16, all the requirements of the Krasnoselskii’s theorem are satisfied. Thus, there exists a fixed point z∈Ω such that z=Y1z+Y2z; this fixed point is a solution of (5). Hence, (5) has a ζ-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 T-periodic solution.Proof.
Let the mappingY be presented by
(49)Yuζ=∫−∞0Qrqζ,uζ+τrdr+∫−∞∞Gζ,ωAω∫−∞0Qrqω,uω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,uω+σ1r,⋯,uω+σmrdrdω.
Foru1,u2∈BCℝ,ℝn, we have
(50)Yu1ζ−Yu2ζ≤∫−∞0Qrqζ,u1ζ+τrdr−∫−∞0Qrqζ,u2ζ+τrdr+∫−∞∞Gζ,ωAω∫−∞0Qrqω,u1ω+τr−qω,u2ω+τrdrdω+∫−∞∞Gζ,ω∫−∞0Qrfω,u1ω+σ1r,⋯,u1ω+σmr−fω,u2ω+σ1r,⋯,u2ω+σmrdrdω=L1+μλL1+L2mu1−u2.
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 qζ,yζ−τζ=cyζ−τ, 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 n-dimentional case.



---

*Source: 6319312-2022-08-01.xml*