INTERNAL LAYERS FOR A QUASI–LINEAR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATION∗

The current paper is mainly concerned with the internal layers for a quasi–linear singularly perturbed differential equation with time delays. By using the method of boundary layer functions and the theory of contrast structures, the existence of a uniformly valid smooth solution is proved, and the asymptotic expansion is constructed. As an application, a concrete example is presented to demonstrate the effectiveness of our result.


Introduction
Delay differential equations (DDEs) arise naturally in just about every interaction of the real world, and the original motivations for studying DDEs mainly come from their application in feedback control theory. In the past several decades, DDEs have been extensively used in control theory [2,3,[5][6][7]13], population dynamics [8,22] and many other scientific fields, where the delays naturally appear to account for a variety of situations. Among them, singularly perturbed DDEs are particularly relevant to describe lots of practical phenomena in many areas, see e.g., [1,4,10,18,19] and the references therein.
The effective method to investigate singularly perturbed DDEs include multiscale method [9], the theory of boundary layer functions established by A. Vasil'eva [21], etc. By the theory of boundary layer functions, Vasil'eva [20] firstly discussed a kind of neutral type equation with small lag wherein τ > 0 is sufficiently small and φ(t) is a given function, and the following results are obtained (a) If f (t, x, y, z) does not depend on z, then for τ → 0, the solution of (1.1) tends to a solution of initial probleṁ x(t)), x(0) = φ(0); (b) Otherwise, then the author declared that the result in (a) is valid if for all (t, x, y, z) in the region defined by φ(0).
With the rapid development of boundary layer functions theory, it has been applied to deal with the related problems of uniformly valid asymptotic solution for Tikhonov systems [11,12,23,24], among which, Ni & Lin [11] have studied a kind of singularly perturbed DDEs with internal layers where 0 < ε ≪ 1 is a small parameter, φ(t) ∈ C([−τ, 0]), τ > 0 is deviation argument, τ < T < 2τ is positive constant. The author proved the existence of uniformly valid asymptotic smooth solutions, and constructed their asymptotic expansion with an internal transition layer.
Motivated by the above discussions, in this paper, we consider the following quasi-linear singularly perturbed DDEs is a smooth function, y T is a given constant and for simplicity, we assume that T ∈ (σ, 2σ]. Furthermore, the functions A, B are assumed to be sufficiently smooth on region wherein I y represents the admissible range of y(t), and σ ∈ (0, T ) divides the domain D into two parts In what follows, the authors are going to apply the boundary layer function method and the theory contrast structures to study (1.2). Throughout this paper, the notations with superscript " ′ " stand for the derivative of functions on the corresponding variable, and we always make the following assumptions.
Remark 1.1. Assumption 1.1 applies the initial-boundary value conditions for each cauchy problems, and the other cases will be discussed in further study.

Assumption 1.2. The following inequalities are guaranteed
In general, α 1 (σ−) ̸ = α 2 (σ+), for the sake of definiteness, assume that α 2 (σ) < α 1 (σ). Our focus is attracted to the study of existence of an asymptotic expansion to a solution of problem (1.2) in the class C 1 [0, T ] ∩ C 2 (0, σ) ∪ C 2 (σ, T ) , with an internal transition layer localized in a neighbourhood of the point t = σ, namely, a solution close to the function α 1 (t) on the left of this neighborhood and the function α 2 (t) on the right of this neighborhood, which varying from α 1 (t) to α 2 (t) swiftly near t = σ, see Fig. 1. It is found from Assumption 1.1 that there is no boundary layers for system (1.2) on the considered domain. Introducing auxiliary system (which is an analog of Tikhonov associated system, ref. [21, p21-27]) to the original problem by making a variable change z = dy dt , then the following auxiliary system is obtained Obviously, the points (α 1,2 (σ), 0) on the phase plane (ỹ,z) are equilibrium points of system (1.3), and the characteristic equations Dividing the first equation in (1.3) by the second equation, the first-order differential equation can be arrived for the functionsz(ỹ). According to (1.4) and (1.5), we find for y ∈ (α 2 (σ), α 1 (σ)), (i) there exists an unstable manifold W u leaving from the point (α 1 (σ), 0) and (ii) there exists a stable manifold W s entering into the point (α 2 (σ), 0) and We suppose that the following condition hold The remainder part of this paper is organized as follows. Sect. 2 is devoted to algorithm of construction for the asymptotic expansions which are established by the method of boundary layer functions and the theory of contrast structures. And the main result for the existence of solutions and estimation of remainder are presented in Sect. 3. Whereafter in Sect. 4, a concrete quasi-linear singularly perturbed delay differential equation and a series of numerical simulations for some given small parameter ε are carried out to demonstrate the effectiveness of our result.

Algorithm of Constructing the Asymptotic Expansions
The asymptotic expansion to a solution of problem (1.2) is constructed separately on D 1,2 (namely, the left problem and the right problem respectively), denote it by y (−) (t, ε) on D 1 and y (+) (t, ε) on D 2 respectively, and the functions y (−) (t, ε) and y (+) (t, ε) are sewn smoothly, as well as their derivatives z (−) := y ′(−) and z (+) := y ′(+) at the point t = σ, the values p(ε) and z(ε) are unknown and will be determined in the process of constructing the asymptotic expansions of solution of problem (1.2) and are sought in the form of expansions in powers of ε in the following 3) and the right problem P (+) : (t, y) ∈ D 2 , 3)-(2.4) for second-order ordinary differential equations are equivalent to the following respective problems for systems of first-order equations

5) and
(2.6) To describe the behavior of solution in a neighborhood of the transition layer point in detail, we introduce a stretched variable τ = t−σ ε , and the asymptotic expansions of the solutions to systems (2.5)-(2.6) are constructed in the following forms represent the regular part of the asymptotic solution (2.7), which mainly reflects the behavior of the solution outside of the internal layers, and denote the internal layer part of the asymptotic solution (2.7). The internal layer functions are required to satisfy Rewriting the sewing conditions (2.1) with regards to the expansions (2.2), (2.8)-

Regular parts
According to the standard Vasil'eva boundary layer functions method, the equations for the terms of the regular part can be obtained by substituting the expressions (2.8) for the functionsȳ (∓) (t, ε) andz (∓) (t, ε) into the system of equations and into the additional conditions at t = 0 and t = T in problems (2.5) and (2.6) respectively, and by equating the coefficients of like powers of ε on both sides of the obtaining equations. Then the functions y (∓) k (t) are determined for each k = 0, 1, · · · as the solutions of Cauchy problems for first-order differential equations, after which the expressions for their derivatives z (∓) k (t) can be found. Indeed, the following Cauchy problem in the zeroth order expansion is obtained There exist solutions of these problems by Assumption 1.1. Takinḡ

determined as the solutions of the problems
. Similarly, the regular termsȳ herein the elements of A(t), ∂ A ∂y (t), ∂ B ∂y (t) are calculated at the point (α 2 (t), α 1 (0), t), and the functions F k (t) and thenz

Internal transition layer parts
The coefficients Q (∓) k y(τ ), k = 0, 1, · · · , and Q (∓) k z(τ ), k = −1, 0, 1, · · · , can be obtained by substituting the expansions (2.9) into the systems of equations (2.12) Equating the coefficients of ε 0 in the second equations in (2.12) and in conditions (2.10) as well as the coefficients of ε −1 in the second equations in (2.12), the following problems for the leading terms of the expansions (2.9) can be obtained and (

2.19)
It is worthy to be noted that p 0 is unknown which will be determined later. Back to systems ( The proof of Lemma 2.1 is similar to the proof in [11], which is omitted here. With regards to the functions Q (−) k y(τ ) and Q (−) k−1 z(τ ), k ≥ 1, can be obtained by the following systems Since That is to say, the functions {Q By Liouville formula and variation of constants formula, it follows that Homoplastically, we have Similarly, we have where C k1 , C k2 , κ > 0 are some positive constants.

Coefficients p k in expansion (2.3)
this actually is Assumption 1.4 with respect to σ and hence we cannot obtain p 0 by G(p) = 0. Thus we pay attention to {Q It is easy to find that f sequentially, we get Similarly, it can be obtained α 1 (0), σ). For the sake of convenience, definẽ 1 (s), s ∈ [0, +∞), according to (2.19), it can be achieved The sewing conditions for determining coefficients p k take the following form d dtȳ in other word, the sewing conditions writē where Ψ is relevant to p i (i = 0, 1, · · · , k − 1), thus Assumption 2.1 implies the existence and uniqueness of p k .

Main Result
This section is devoted to the existence of solutions and estimation of remainder, the main result is presented as follows.

Application
A concrete example is presented to demonstrate the effectiveness of the main result.
Example 4.1. Consider a second-order quasi-linear singularly perturbed delay differential equation in the following form (0 < ε ≪ 1) In what follows, the first-order smooth asymptotic solution with internal transition layers are constructed and the effects of parameter ε are shown in several figures by choosing some fixed values of ε.
By the method of steps, it can be found where α 1,2 (t) are solutions of Cauchy problems in Assumption 1.1. It is found that there takes place an internal transition phenomenon for system (4.1) on the interval [0,2] (See Fig. 2) and easy to verify that system (4.1) satisfy Assumptions 1.  Obviously, H(p) = 0 has a solution p = 0 and dH(0) dp = −1 ̸ = 0, which means that Assumption 2.1 is fulfilled well.
Introducing a stretched variable τ = (t−1)/ε, according to Theorem 3.1, system (4.1) has a uniformly valid asymptotic solution y(t, ε) =  In the following, we give the graphics of the first-order smooth asymptotic solution y 1 (t, ε) for some given ε, see Fig. 3.