含积分算子拟线性系统边值问题的奇摄动

The Singular Perturbation of Boundary Value Problem of Quasi-Linear System with Integral Operator

研究奇摄动积分微分方程组边值问题εy″=f(x,y,Ty,ε) y′+g(x,y,Ty,ε) ; y(0 ,ε) =A(ε) , y(1 ,ε) =B(ε)其中 y、g、A和 B均为 n维向量函数 ,f是 n× n矩阵函数 ,(Ty) (x) =∫x0 K(x,s,y(s,ε) ,ε) ds.在一定假设条件下 ,利用对角化技巧和逐步逼近法证明解的存在 ,并给出解的直到 O(εN + 1 )的渐近展开式 .

In this paper, we study the singularly perturbed system of integral differential equations: εy″=f(x,y,Ty,ε)y′+g(x,y,Ty,ε), y(0,ε)=A(ε), y(1,ε)=B(ε) where y、g、A and B belong to R n, (Ty)(x)=∫ x 0K(x,s,y(s,ε),ε) d s, f is an n×n matrix function. Under the appropriate conditions, we prove the existence of the solution and give its asymptotic expansions till O(ε N+1 ) by the method of diagonalization and successive approximation.