Study On Singularly Perturbed Problems With Integral Boundary Condition

Singular perturbation problem is an old and yet novel topic. Because of its wide application in various fields, it has drawn much attention of investigators.Different equations with integral boundary condition have applications in various physical problems. Over the last decades, some existence and uniqueness results for problems with integral boundary conditions have been obtained, and various numerical methods have been developed. This paper mainly deals with the singular perturbation problems with integral boundary condition by boundary layer function method.First, we study the singular perturbed first-order differential equations with integral boundary condition as follows:where 0 <ε<<1,μand d are given constants. f(t,x) is sufficiently continuously dif-ferentiable in I×R, b(t) is continuous in [0,1]. Asymptotic solution was constructed by means of boundary layer function method. And then, we prove the existence of the solution of this singular perturbed problem and the estimation of the remainder term with Banach fixed point theorem. And then,we discuss linear second-order singularly perturbed problem with integral boundary condition as follows:and the Dirichlet problem for semi-linear second-order singularly perturbed equation with integral boundary condition as follows:where f : [0,1]×R→R is sufficiently continuously differentiable and h_i,g(t) andf(t) is continuous. By the method of boundary layer function the formal asymptotic solution was constructed, and its uniformly validity was proved using the theory of differential inequalities.At last, we discuss the Robin problem for nonlinear second-order singularly perturbed equation with integral boundary condition as follows:where f: [0,1]×R~2→R is sufficiently continuously differentiable and h_i : R→R is continuous, k_i is positive constant (i = 1,2).Generated theory of differential inequalities of nonlinear second-order singularly perturbed equation with integral boundary condition was obtained from the theory of differential inequalities of semi-linear second-order singularly perturbed equation with integral boundary condition,and then uniformly validity of asymptotic solution of the original problem was proved.