The step-type contrast structure for a second order semi-linear singularly perturbed differential-difference equation

The step-type contrast structure for a second order semi-linear singularly perturbed differential-difference equation is studied. Using the methods of boundary function and fractional steps, we construct the formula asymptotic expansion of the problem. At the same time, based on sewing techniques, the existence of the step-type contrast structure solution and the uniform validity of the asymptotic expansion are proved.


Introduction
The boundary-value problems for singularly perturbed differential-difference equations arise in various practical problems in biomechanics and physics such as in variational problem in control theory and depolarization in Stein's model. Many scholars have done a lot of work on this field, especially for linear problems [1][2][3][4][5][6]. For nonlinear problems, some results [7][8][9][10][11][12] have also been obtained. However, most of these works are related to boundary layers, numerical solution, or the proof of the existence of the solution. Few of them concern the contrast structures and the uniform validity of the asymptotic expansions [7,12]. Recently, the contrast structures have become the focus of attention in singular perturbation [13][14][15][16]. The fundamental characteristic of contrast structures is that there exists a t * (or multiple t * ) within the domain of interest, which is called an internal transition point. The position of t * is unknown in advance, and it needs to be determined thereafter. In the neighborhood of t * , the solution y(t, μ) will have an abrupt structure change. In the different sides of t * , if y(t, μ) approaches different reduced solutions, we call it step-type contrast structure. If y(t, μ) approaches the same reduced solution, we call it spike-type contrast structure. In [17], Wang, Xu, and Ni study the spike-type contrast structure for the following singularly perturbed differential-difference equation which only contains negative shift in it: μ 2 y (t) = F y(t), y(tσ ), t , 0≤ t ≤ T; (1.1) In this paper, we study the step-type contrast structure for system (1.1), (1.2), where 0 < μ 1 is a small parameter and σ is a delay argument. α(t) is a smooth function defined in [-σ , 0]. T is a positive constant that satisfies σ ≤ T ≤ 2σ . The restriction on T will not influence the essence of the problem and it is only convenient for our discussion.

Algorithm for the construction of asymptotics
Let μy = z, then (1.1) can be rewritten as follows: When necessary we impose several additional conditions on Eq. (2.1).
For k x(τ 0 ), we have the following system: whereF y gets its value at the point (ϕ 1 According to the Liouville formulas and the constant-change method, we infer that (2.13) Thus, k x(τ 0 ) are completely determined. The exponential decay of k x(τ 0 ) can easily be obtained from (2.13).
For boundary functions R k x(τ T ) (k ≥ 0), they have no essential influence on the interior layer and the solving method of them completely coincides with k x(τ 0 ) (k ≥ 0). We will not discuss them in detail.
which is the equation for finding t 0 .
(3.11) (H 5 ) Suppose that there exists a solution p 0 =p 0 for (3.11) that satisfies dG dp 0 | p 0 =p 0 < 0. For p k , by virtue of (2.38) and (3.8), we have By (H 5 ), the coefficient of p k is not equal to zero , so p k is determined. ThusQ (±) k x(τ σ ) are all completely determined.

The existence of the complex solution
In this section, using the method of sewing connection, we prove the existence of the solution about problem (2.1), (1.2) and give out the estimates of the remainder. The solution of (2.1), (1.2) may be considered as a solution which is smoothly connected by the solutions of the following auxiliary problems.