The Asymptotic Solutions for a Class of Nonlinear Singular Perturbed Differential Systems with Time delays

We study a kind of vector singular perturbed delay-differential equations. By using the methods of boundary function and fractional steps, we construct the formula of asymptotic expansion and confirm the interior layer at t = σ. Meanwhile, on the basis of functional analysis skill, the existence of the smooth solution and the uniform validity of the asymptotic expansion are proved.


Introduction
Singular perturbed differential equations are often used as mathematical models describing processes in biological sciences and physics, such as genetic engineering and the El Nino phenomenon of atmospheric physics [1]. In order to study natural and social processes more accurately, we often construct the models with small delay time and obtain much behavior of corresponding objects. The models are mostly expressed by singular perturbed delay-differential equations. So, singular perturbed delay-differential equations can express the processes more exactly. Studying the singular perturbed delay-differential problem is a very attractive object in the mathematical circle.
In addition, in the study of population models and propagation of epidemic virus, we sometimes require the construction of models. The models are often expressed by singular perturbed delay-differential equations. We can get the equilibrium points of singular perturbed delaydifferential equations and confirm the laws of processes. Therefore, using the research methods and theoretical results of singular perturbed delay-differential problems to solve natural and social processes is essential.
In recent years, more and more attention was paid to the study of singular perturbed delay-differential problems [2], especially for scalar boundary value problems [3][4][5], but vector boundary value problems are rarely seen [6,7]. Up to now, the vector theory of singular perturbed problems is still not mature. Wang and Ni study a class of semilinear singularly perturbed equations using the method of fractional steps [8]. By the method of boundary layer function [9], Wang considered a kind of nonlinear singularly perturbed boundary value problems [10].
In this paper, we will discuss the interior layer for a class of nonlinear singularly perturbed differential-difference equations and construct its asymptotic expansion formula. Then, the existence of the smooth interior layer solution and the uniform validity of the asymptotic expansion are proved. The results of this paper are new and complement the previous ones.
We consider the following nonlinear singularly perturbed differential-difference equations where The restriction on 2 will not influence the essence of the problems.
We will use the method of fractional steps to discuss the system (1). Let = 0; then we can obtain the degenerate equations (3) and (4) of (1) The degenerate problem (3) is solvable. We hypothesize that the solution of system (3) is y (1) ( ) = ( ); substituting y (1) ( ) = ( ) into (4) yields y (2) ( ) = ( ). Thus, we have the degenerate solution y( ) on the interval [0, 2 ], namely, the following: According to the truth of boundary layer functions and interior layer functions in [3], we can confirm that the interior layer may occur at = and boundary layers may occur at the two terminal points of interval [0, 2 ].

Lemma 2. Under the condition (H4), the boundary functions
x( ) satisfy the following inequality: where 2 , 2 are positive constants.

Lemma 3. Under conditions (H5)-(H7), the boundary functions
Consider the first approximate system of (47) In the same way, we can confirm that there exists andimensional stable manifold, which is in some region 4 of vector function 0 y.
By (H8), the system (47) has a solution. 0 y and 0 z are both satisfied with exponential decay estimate.
For x, we have whereF ,F take value at ( 0 z, y 0 (2 ) + 0 y, ( ), 2 ) and ( 0 ) is a known vector function. The determination of x( * ) is treated in the same way as Π x( 0 ) and is omitted here. Obviously, x( * ) satisfy the following Lemma.

Lemma 4. Under the condition (H8), the boundary functions
x( * ) satisfy the following inequality: where 4 , 4 are all positive constants.

The Main Result
Let

Example
Let us consider the systems where y = ( 1 , 2 ) . For 0 ≤ ≤ 1, the degenerate solution of (58) is Obviously, conditions (H2) and (H3) hold. Π 0 y( 0 ), 0 y( * ), The Scientific World Journal 7 We can see that the zero order asymptotic solution is close to the reduced solution.

Conclusive Remarks
Using the boundary layer function method, we consider a class of -dimensional singularly perturbed differential equations with time delay. Under some assumptions, we obtain the asymptotic solution of the system (1). In comparison with [7] an [9], the system we study is more general. We use functional method to solve the asymptotic solution of (1) in this paper. It is different from the numerical solution obtained by numerical method. The asymptotic solution of (1) can be used in analytic calculation and obtain the asymptotic behaviors for deeper physical quantities.