# Response Solutions for a Singularly Perturbed System Involving Reflection of the Argument

Authors: Shanliang Zhu; Shufang Zhang; Xinli Zhang; Qingling Li
Journal: Advances in Mathematical Physics
 (2020)

---

## Abstract

In this paper, the existence and uniqueness of response solutions, which has the same frequencyω with the nonlinear terms, are investigated for a quasiperiodic singularly perturbed system involving reflection of the argument. Firstly, we prove that all quasiperiodic functions with the frequency ω form a Banach space. Then, we obtain the existence and uniqueness of quasiperiodic solutions by means of the fixed-point methods and the B-property of quasiperiodic functions.


---

## Body

## 1. Introduction

The following singularly perturbed system(1)x′t=Ft,xt,yt,ε,εy′t=Gt,xt,yt,ε,occurs in many areas, including biochemical kinetics, genetics, plasma physics, and mechanical and electrical systems involving large damping or resistance [1–4], where x and y are vectors with multiple components and ε≥0 is a small parameter. The existence of periodic solutions and almost periodic solutions of (1) had been one of the most attracting topics in the qualitative theory of ordinary differential equations. The early contributions on these topics are due to Anosov [5] and Flatto and Levinson [6]. They investigated system (1) in the case that the degenerate system
(2)x′t=Ft,xt,yt,0,0=Gt,xt,yt,0,has a periodic solution θt,χt. The authors showed sufficient conditions on F,G which assure that the existence of periodic solutions of (1) and these solutions converge to θt,χt as ε⟶0 uniformly. In 1961, Hale and Seifert [7] generalized the results of Flatto and Levinson to the almost periodic case and gave sufficient conditions for the existence of the almost periodic solutions of (1) using the similar method with [6]. Chang [8] obtained the same result of [7] under generalized hypothesis. But, the above papers [5–8] do not consider the stability properties of the solutions.Smith [9] considered the existence of almost periodic or periodic solutions for system (1). By the construction of manifolds of initial data, the author investigated the stability properties of these solutions, which approach the given solutions as t⟶∞ at an exponential rate, α, independent of ε. He also gave the application in a reaction diffusion system with a traveling wave input.It is natural to ask whether there is a bounded solution of system (1) for sufficiently small ε and how the stability properties of the solutions for the quasiperiodic case are.For the Silberstein equation(3)x′t=x1t,we define yt=xet, then Equation (3) is equivalent to y′t=e−ty−t, which is known as the equation involving reflection of argument. This kind of equations has applications in the study of stability of differential-difference equations, see Sharkovskii [10]. One of the earliest contributions to this kind of equations are due to Wiener and Aftabizadeh [11]. They investigated the boundary value problems for the second-order nonlinear differential equation
(4)y″t=ft,yt,y−t,y−a=y0,ya=y1by Schauder fixed-point theorem, where f∈C−a,a×ℝ×ℝ,ℝ. They also considered the boundary value problems for the following equation
(5)y″t=ft,yt,y−t,y′−a−hy−a=0,y′a+kya=0,by changing the equation to a higher order one without reflection of the argument, where h,k≥0,h+k>0. Gupta [12, 13] studied more general boundary value problems than Equations (4) and (5) using degree theory arguments. He proved the existence of solutions for the boundary value problems in a simple and straightforward manner. The existence and uniqueness of periodic, almost periodic, pseudo almost periodic, Besicovitch almost periodic, and pseudo almost automorphic solutions of this kind of equations were investigated in [14–19]. Cabada et al. [20–22] studied the first-order equation with two-point boundary conditions and the nth-order differential equations involving reflection, constant coefficients, and initial conditions, adding a new element to the previous studies: the existence of Green’s function.However, as far as we know, the quasiperiodic solutions for the equations involving reflection of the argument have not been considered yet. Our present paper is devoted to discuss the existence and uniqueness of response solutions for the following singularly perturbed system(6)x′t=Ft,xt,x−t,yt,y−t,ε,εy′t=Gt,xt,x−t,yt,y−t,ε,where ε≥0 is a small real parameter, and the functions F,G are quasiperiodic in t uniformly on ℝ2×ℝ2 with frequency ω=ω1,ω2,⋯,ωd. A quasiperiodic solution of (6) with the frequency ω is called response solution.It is assumed that the degenerate system(7)x′t=Ft,xt,x−t,yt,y−t,0,0=Gt,xt,x−t,yt,y−t,0,has a quasiperiodic “outer” solution which we take to be the trivial solution, that is, we suppose
(8)Ft,0,0,0,0,0≡Gt,0,0,0,0,0≡0so that x,y=0,0 satisfies (7). Expanding (6) about the trivial solution gives
(9)x′t=a1t,εxt+a2t,εx−t+b1t,εyt+b2t,εy−t+ft,xt,x−t,yt,y−t,ε,εy′t=c1t,εxt+c2t,εx−t+d1t,εyt+d2t,εy−t+gt,xt,x−t,yt,y−t,ε.One can think of, e.g.,a1t,ε=∂F/∂xt,0,0,0,0,ε. In the following discussion, we mainly consider (9).This paper is organized as follows: in Section2, we present the Bohr’s notion of B-property for quasiperiodic functions and then prove that all ω-frequency continuous quasiperiodic functions form a Banach space under the supremum norm. We prove an existence and uniqueness result for a linear scalar equation with reflection of the argument. In Section 3, the main results on the local existence and uniqueness of response solutions will be stated and proved by means of fixed-point methods in the spirit of Sacker and Sell [23]. We give conclusions of this paper in Section 4.


## 2. Preliminary

Firstly, we will give some lemmas which are important in proving our main results.Definition 1.
Assume thatω1,ω2,⋯,ωd∈ℝ are rationally independent. A continuous function ut on ℝ is said to be quasiperiodic with frequencies ω1,ω2,⋯,ωd, if there exists a periodic function U=Uθ1,θ2,⋯,θd (called the lift of u) in θ1,θ2,⋯,θd with the same period 2π, such that
(10)ut=Uω1t,ω2t,⋯,ωdt,∀t∈ℝ.Remark 2.
This definition for quasiperiodic function can be found in many references, for example [24]. It is not difficult to prove that this definition is equivalent to the definition of quasiperiodic function in [25].LetQPω be the set of all quasiperiodic functions with frequency ω=ω1,ω2,⋯,ωd.Definition 3. (see [4]).
A functionHt:ℝ⟶ℝ is said to have a B-property on a set of real numbers ω1,ω2,⋯,ωd, if
(i)
H is continuous on ℝ
(ii)
for everyε>0, there is δ=δε>0 such that if a real number τ satisfies the d Diophantine inequalities
(11)ωkτ≤δmod2πk=1,⋯,d,then, τ is an ε-translation number of H, i.e.,
(12)supt∈ℝHt+τ−Ht≤ε.Lemma 4. (see [3]).
Suppose thatHt is a quasiperiodic function with frequencies ω1,ω2,⋯,ωd. Then, Ht has the B-property on ω1,ω2,⋯,ωd, Conversely, if Ht has the B-property on a finite rationally independent set ω1,ω2,⋯,ωd, then Ht is a quasiperiodic function with frequencies contained in ω1,ω2,⋯,ωd.Proof.
The proof of the lemma can be found in [26].Lemma 5.
QPω,. is a Banach space with the norm f=supftt∈ℝ.Proof.
Suppose thatfnt∈QPω,n=1,2,⋯ is a Cauchy sequence. By the fact that QPω is a subspace of CBℝ, which is a Banach space of bounded continuous function on ℝ with norm f=supt∈ℝft, there is a f0t∈CBℝ such that fn−f0⟶0 (as n⟶∞). So for any ε>0 and all t∈ℝ, there exists a K∈ℕ, such that fKt−f0t≤ε/3.
SincefKt∈QPω, fKt has the B-property on ω1,ω2,⋯,ωd by Lemma 4. So for the ε, there is a δ>0, such that if a real number τ satisfies the inequalities ωkτ≤δmod2πk=1,⋯,d, then we have fKt+τ−fKt≤ε/3 for all t∈ℝ. Furthermore, we have
(13)f0t+τ−f0t≤fKt+τ−f0t+τ+fKt+τ−fKt+fKt−f0t≤ε3+ε3+ε3=ε,thus, f0t has the B-property on ω1,ω2,⋯,ωd. Therefore, f0t is quasiperiodic, i.e., f0t∈QPω.Corollary 6.
QPω2≔QPω×QPω is a Banach space with the norm f,g≔f+g.Lemma 7.
IfHt∈QPω, then H−t∈QPω.Proof.
SinceHt∈QPω, Ht has the B-property on the set ω1,ω2,⋯,ωd by Lemma 4. Then for every ε>0, there exists a τ, which satisfies (11), is an ε-translation number of H. For these τ, we have
(14)supt∈ℝH−t+τ−H−t≤sups∈ℝHs−Hs+τ=sups∈ℝHs+τ−Hs≤ε.
SoH−t has the B-property on the set ω1,ω2,⋯,ωd. Therefore, H−t∈QPω by Lemma 4.Lemma 8.
There existε0>0 and L1>0 such that for each ht∈QPω, λ2=α2−β2/ε2, λ>0, the equation
(15)εz′t=αzt+βz−t+ht,where β≠0, has a unique solution zh,εt∈QPω for 0<ε<ε0. Moreover, the operator Lε1:h⟶zh,ε is linear and satisfies ∥Lε1∥≤L1. Furthermore, the map ε⟶Lε1 is continuous for 0<ε≤ε0.Proof.
Existence. Similar to the proof of Lemma 2 in [14], we can verify that
(16)zh,εt=−12λeλt∫t∞e−λsλ+αεhsε−βh−sε2ds+12λe−λt∫−∞teλsλ−αεhsε+βh−sε2ds,is a particular solution of Equation (15) for any ht∈QPω. Now, we show zh,εt∈QPω.
Sinceht∈QPω, ht,h−t has the B-property on the set ω1,ω2,⋯,ωd by Lemma 4 and Lemma 7. Then for every ε>0, if τ satisfies inequality (11), it will be an ε-translation number of ht and h−t. For this τ, we have
(17)zh,εt+τ−zh,εt=−12λeλt∫t∞e−λsλ+αεhs+τ−hsε−βh−s−τ−h−sε2ds+12λe−λt∫−∞teλsλ−αεhs+τ−hsε+βh−s−τ−h−sε2ds≤12λeλt∫t∞e−λsλ+αεεε+βεε2ds+12λe−λt∫−∞teλsλ−αεεε+βεε2ds≤ελ−α+ελ+α+2βε2α2−β2.
Sozh,εt has the B-property on the set ω1,ω2,⋯,ωd for 0<ε<ε0. Hence, zh,εt∈QPω.
Uniqueness. If there was another quasiperiodic solutionz~h,εt for Equation (15), then the difference ut=zh,εt−z~h,εt should be a solution of the homogeneous equation
(18)εu′t=αut+βu−t.
According to the Lemma 2 of [14], we see that ut is of the form
(19)ut=Cελ−αβe−λt+eλt,t∈ℝ,for some constant C. If C≠0, then ut will be unbounded. This is a contradiction to the boundedness of quasiperiodic function.
So, the operatorLε1:h⟶zh,ε is well defined. From (16), we see the operator Lε1 is linear. On the other hand,
(20)zh,εt≤M1ε0λ−α+ε0λ+α+2β2α2−β2,where M1=∥h∥. So Lε1 satisfies ∥Lε1∥≤L1 with L1=ε0λ−α+ε0λ+α+2β/2α2−β2.
To prove the continuity ofLε1 in ε, we write vt=zh,ε1t−zh,ε2t for any 0<ε1,ε2<ε0, then vt satisfies
(21)ε1v′t=αvt+βv−t+ε2−ε1ε2αzh,ε2t+βzh,ε2−t+ht.
It follows that(22)vt=zh,ε1t−zh,ε2t≤ε2−ε1ε2ε0λ−α+ε0λ+α+2β2α2−β2αzh,ε2t+βzh,ε2t+ht≤ε2−ε1ε2ε0λ−α+ε0λ+α+2β2α2−β2αL1+βL1+1M1
This implies that the mapε⟶Lε1 is continuous for 0<ε≤ε0.Similar to the proof of Lemma8, one can prove the following Lemma.Lemma 9.
There existsL2>0 such that for each rt∈QPω, γ2=α2−β2, γ>0, the equation
(23)w′t=αwt+βw−t+rt,has a unique solution wrt∈QPω for β≠0. The map r⟶wr defines a bounded linear operator satisfying ∥w∥≤L2∥r∥.For the sake of convenience, we state the following conditions.(H1) ait,ε,bit,ε,cit,ε,dit,ε∈QPω,i=1,2 are continuous in ε, uniformly in t∈ℝ. Let M2 denote a common bound for these functions on t,ε∈ℝ×0,ε0.(H2) ait,0=ai0,bit,0=bi0,cit,0=0,dit,0=di0,i=1,2 are constants and d20≠0, a20≠0. Moreover, a102−a202>0,d102−d202>0.(H3) The functions f,g are quasiperiodic in t uniformly on x1,x2,y1,y2 such that t∈ℝ,xi,∣yi∣≤σ0i=1,2,0≤ε≤ε0,0≤σ≤σ0. Moreover, there are two nondecreasing functions Φε,Ψε,σ, which satisfy
(24)limε→0Φε=0,limε,σ→0,0Ψε,σ=0such that
(25)ft,0,0,0,0,ε≤Φε,gt,0,0,0,0,ε≤Φε,t∈ℝ,0≤ε≤ε0,∣ft,x1,x2,y1,y2,ε−ft,x^1,x^2,y^1,y^2,ε∣≤Ψε,σ∑i=12∣xi−x^i∣+∣yi−y^i∣,∣gt,x1,x2,y1,y2,ε−gt,x^1,x^2,y^1,y^2,ε∣≤Ψε,σ∑i=12∣xi−x^i∣+∣yi−y^i∣hold for all t∈ℝ, i f xi,x^i,yi,y^i≤σ, 0≤ε≤ε0, 0≤σ≤σ0.


## 3. Main Results

First, we consider the following linear system:(26)x′t=a1t,εxt+a2t,εx−t+b1t,εyt+b2t,εy−t+f~t,εy′t=c1t,εxt+c2t,εx−t+d1t,εyt+d2t,εy−t+g~t,where f~, g~∈QPω.Theorem 10.
If (H1)–(H2) hold. Then there exist ε1, 0<ε1≤ε0, positive functions Ai,jε,1≤i,j≤2 defined for 0<ε≤ε1, satisfying
(27)limε→0+A1,1ε=L2,limε→0+A1,2ε=2L1L2M2,limε→0+A2,1ε=0,limε→0+A2,2ε=L1,Ai,jε≤2L1L2M2+L1+L2,1≤i,j≤2,such that for each f~,g~∈QPω2, 0<ε≤ε1, system (26) has a unique solution xf~,g~,εt,yf~,g~,εt∈QPω2 which satisfies
(28)x≤A1,1εf~+A1,2εg~,y≤A2,1εf~+A2,2εg~.
The mapf~,g~⟶xf~,g~,ε,yf~,g~,ε defines a bounded linear operator Kε satisfying Kε≤2L1L2M2+L1+L2 and ε⟶Kε is continuous for 0<ε≤ε1.Proof.
Givenf~,g~∈QPω2, x0t,y0t∈QPω2. Define xt,yt as the solution of the system
(29)x′t=a10xt+a20x−t+a1t,ε−a10x0t+a2t,ε−a20x0−t+b1t,εyt+b2t,εy−t+f~t,εy′t=d10yt+d20y−t+d1t,ε−d10y0t+d2t,ε−d20y0−t+c1t,εx0t+c2t,εx0−t+g~t.
The second equation in (29) has a unique solution y∈QPω by (H1), (H2) and Lemma 8. Then, put this y into the first equation which is solved for a unique x∈QPω using Lemma 9. Writing x,y=T1x0,y0,f~,g~,ε, then solving (26) is equivalent to finding a fixed point of T1·,·,f~,g~,ε
Ifxi,yi=T1x0i,y0i,f~,g~,ε,i=1,2, then we find that u=x1−x2,v=y1−y2 satisfy
(30)u′t=a10ut+a20u−t+a1t,ε−a10x01t−x02t+a2t,ε−a20x01−t−x02−t+b1t,εvt+b2t,εv−t,εv′t=d10vt+d20v−t+d1t,ε−d10y01t−y02t+d2t,ε−d20y01−t−y02−t+c1t,εx01t−x02t+c2t,εx01−t−x02−t.
From Lemma8, Lemma 9 and (H1), it follows that
(31)u≤L2a1t,ε−a10+a2t,ε−a20x01−x02+2M2v,v≤L1c1t,ε+c2t,εx01−x02+d1t,ε−d10+d2t,ε−d20y01−y02.
And this leads to the estimate(32)u≤L2a1t,ε−a10+a2t,ε−a20x01−x02+2M2v,≤L2a1t,ε−a10+a2t,ε−a20x01−x02+2L1L2M2c1t,ε+c2t,εx01−x02+2L1L2M2d1t,ε−d10+d2t,ε−d20y01−y02.
From the hypothesis (H1) and (H2), it follows that there exists ε1≤ε0 such that
(33)L2a1t,ε−a10+a2t,ε−a20+2L1L2M2c1t,ε+c2t,ε<12,2L1L2M2d1t,ε−d10+d2t,ε−d20<12,L1c1t,ε+c2t,ε<12,L1d1t,ε−d10+d2t,ε−d20<12for 0<ε≤ε0. The contraction mapping principle implies that T1 has a unique fixed point x∗,y∗∈QPω2. It follows from (29), Lemma 8, and Lemma 9 that
(34)x∗≤L2a1t,ε−a10+a2t,ε−a20x∗+2M2y∗+f~,y∗≤L1c1t,ε+c2t,εx∗+d1t,ε−d10+d2t,ε−d20y∗+g~,which imply
(35)x∗≤1−L2a1t,ε−a10+a2t,ε−a20−12L2M2y∗+L2f~,y∗≤1−L1d1t,ε−d10+d2t,ε−d20−1L1c1t,ε+c2t,εx∗+L1g~.
Putting the second inequality of (35) into the first gives
(36)x∗≤A1,1εf~+A1,2εg~,where
(37)A1,1ε=1−2pεqεL1L2M2c1t,ε+c2t,ε−1pεL2,A1,2ε=21−2pεqεL1L2M2c1t,ε+c2t,ε−1pεqεL1L2M2,pε=1−L2a1t,ε−a10+a2t,ε−a20−1,qε=1−L1d1t,ε−d10+d2t,ε−d20−1.
Putting (36) into the second inequality of (35) gives
(38)y∗≤A2,1εf~+A2,2εg~,where
(39)A2,1ε=A1,1εqεL1c1t,ε+c2t,ε,A2,2ε=A1,2εqεL1c1t,ε+c2t,ε+qεL1.
The linear operatorKε:f~,g~⟶x∗,y∗ is bounded with
(40)x∗+y∗≤A1,1ε+A2,1εf~+A1,2ε+A2,2εg~,≤22L1L2M2+L1+L2f~+g~provided that ε1 is so small that Ai,jε≤2L1L2M2+L1+L2 for 0<ε≤ε1. Thus, Kε≤22L1L2M2+L1+L2.
Now, we consider the continuity of the mapε⟶Kε. If we write xt,ε=xf~,g~,εt for 0<ε≤ε1, then
(41)ut=xt,ε1−xt,ε2,vt=yt,ε1−yt,ε2satisfy
(42)u′t=a1t,ε1ut+a2t,ε1u−t+b1t,ε1vt+b2t,ε1v−t+a1t,ε1−a1t,ε2xt,ε2+a2t,ε1−a2t,ε2x−t,ε2+b1t,ε1−b1t,ε2yt,ε2+b2t,ε1−b2t,ε2y−t,ε2,ε1v′t=c1t,ε1ut+c2t,ε1u−t+d1t,ε1vt+d2t,ε1v−t+c1t,ε1−c1t,ε2xt,ε2+c2t,ε1−c2t,ε2x−t,ε2+d1t,ε1−d1t,ε2yt,ε2+d2t,ε1−d2t,ε2y−t,ε2+ε2−ε1ε2c1t,ε2xt,ε2+c2t,ε2x−t,ε2+d1t,ε2yt,ε2+d2t,ε2y−t,ε2+g~t.
In view of (36), (38), and (40), it follows that the map ε⟶Kε is continuous for 0<ε≤ε1.We now consider the nonlinear system (9)
(43)x′t=a1t,εxt+a2t,εx−t+b1t,εyt+b2t,εy−t+ft,xt,x−t,yt,y−t,ε,εy′t=c1t,εxt+c2t,εx−t+d1t,εyt+d2t,εy−t+gt,xt,x−t,yt,y−t,ε.Theorem 11.
Suppose that (H1), (H2) and (H3) hold. Then there exist 0<ε2≤ε1,0<σ1≤σ0 such that for each ε satisfying 0<ε≤ε2 and f,g∈QPω2, system (9) has a unique solution xt,ε,yt,ε∈QPω2, which satisfies
(44)x≤σ1,y≤σ1,x+y=OΦ∈,∈⟶0,and is continuous in ε uniformly for t∈ℝ.Proof.
From (H3), we can choose σ1 and ε2 such that
(45)22L1L2M2+L1+L24σ1Ψε2,σ+Φε2<σ1,42L1L2M2+L1+L2Ψε2,σ≤12.
For anyx0,y0∈QPω2 with x0≤σ1,y0≤σ1,0<ε≤ε2, consider the system
(46)x′t=a1t,εxt+a2t,εx−t+b1t,εyt+b2t,εy−t+ft,x0t,x0−t,y0t,y0−t,ε,εy′t=c1t,εxt+c2t,εx−t+d1t,εyt+d2t,εy−t+gt,x0t,x0−t,y0t,y0−t,ε.
By Theorem10, system (46) has a unique solution x,y∈QPω2 and the estimate
(47)ft,x0t,x0−t,y0t,y0−t,ε≤Ψε,σ2x0+y0+Φε≤4σ1Ψε2,σ1+Φε2.
Writingx,y=Lε2f·,x0,y0,ε,g·,x0,y0,ε≡T2x0,y0,ε, then the existence of a solution of (9) is equivalent to the existence of a fixed point of the mapping T2. We may estimate x,y by Theorem 10
(48)x≤A1,1ε+A1,2ε4σ1Ψε2,σ1+Φε2,≤22L1L2M2+L1+L24σ1Ψε2,σ1+Φε2,and similarly for ∥y∥. If xi,yi=T2x0i,y0i,ε,i=1,2, then we find that
(49)x1−x2≤22L1L2M2+L1+L2Ψε,σx01−x02+y01−y02and similarly for y. We conclude
(50)x1−x2+y1−y2≤42L1L2M2+L1+L2Ψε2,σx01−x02+y01−y02.
Hence, the mappingT2·,·,ε maps the closed set Ω=x0,y0∈QPω2:x0≤σ1,y0≤σ1 into itself for each ε with 0<ε≤ε2 and is a uniform contraction in view of (45), (48), and (50).
It follows thatLε2 is continuous since f,g are continuous in x,y,ε uniformly for t. For fixed x0,y0∈QPω2, the map ε⟶T2x0,y0,ε is continuous on 0,ε2. It follows from the uniform contraction principle that T2 has a unique fixed point x∗,y∗∈QPω2 which is a continuous function of ε with 0<ε≤ε2.
Finally, we obtain the estimation ofx∗,y∗ from the defining system as
(51)x∗+y∗≤22L1L2M2+L1+L22x∗+y∗Ψε2,σ1+Φε≤12x∗+y∗+22L1L2M2+L1+L2Φε.
So(52)x∗+y∗≤42L1L2M2+L1+L2Φε.In order to explain the practical application of the system proposed in this paper, we consider the following singularly perturbed equations, which is closely related to a class of equations widely applied in the field of engineering technology and wave theory of physics.Example 1. (Practical example).
Consider a class of singularly perturbed equations, which can be described as follows:(53)εU″s−U′s−U′1s1s2−hs,Us,U1s,ε=0.
Introducing the variables(54)Us=U0s+us,where U0s is the bounded solution of the system,
(55)−U′s−U′1s1s2−hs,Us,U1s,0=0,and U0′′s exists. Then us satisfies
(56)εu″s−u′s−u′1s1s2−As,εus−Bs,εu1s=Rs,us,u1s,ε,where
(57)As,ε=h′2s,U0s,U01s,ε,Bs,ε=h′3s,U0s,U01s,εu1s,Rs,us,u1s,ε=−As,εus−Bs,εu1s−hs,U0s,U01s,0+hs,U0s+us,U01s1s+u1s,ε−εU′′0s.
Subsequently, settingXs=εu′s−us+ku1/s,Ys=εu′s, then Equation (56) is equivalent to the system
(58)X′s=As,εXs−Ys+Bs,εX1s−Y1s+Rs,Xs−Ys,X1s−Y1s,ε,εY′s=Ys+Y1s1s2+εAs,εXs−Ys+εBs,εX1s−Y1s+εRs,Xs−Ys,X1s−Y1s,ε.
Finally, the substitutionss=et,Xs=Xet≜xt,Ys=yet≜yt, transform (58) into
(59)x′s=etAet,εxt−yt+Bet,εx−t−y−t+Ret,xt−yt,x−t−y−t,ε,εy′s=etyt+e−ty−t+εetAet,εxt−yt+Bet,εx−t−y−t+Ret,xt−yt,x−t−y−t,ε,which is a form of (9).


## 4. Conclusions

In this paper, we consider the existence of a response solution for a singularly perturbed system involving reflection of the argument. Firstly, we prove that allω-frequency continuous quasiperiodic functions form a Banach space under the supremum norm using the key lemma, that is, Lemma 4. Then, we obtain an existence and uniqueness result for a linear scalar equation with reflection of the argument. Expanding (6) about the trivial solution gives system (9), we firstly prove the existence and uniqueness of response solutions for a linear system (26). Then, we obtain the existence of response solutions for system (9) by means of fixed-point methods.



---

*Source: 6738247-2020-03-21.xml*