| The singular perturbation theory is an unceasingly developing and extremely vital subject. Various singular perturbation methods and theories, such as boundary func-tions, matching, differential inequality theory, geometric theory and so on, have been widely applied in many kinds of science and engineering fields, which play a very im-portant role in solving physical problems. Actually, many kinetic mathematical models possess small parameters. Under most situations, due to the difficulty in obtaining ex-act solutions of complex equations, it is very important to derive the uniformly valid asymptotic solutions by singular perturbation methods. In fact, this kind of asymp-totic solution is a form between precise one and numerical one, which is not only easy for theoretic analysis but also for numerical simulation.Experiencing the development in 1 century, the singular perturbation theory has had rich contents. At present, contrast structures, including step-type and spike-type, have recently been one of the hot topics in singular perturbation problems. This paper is devoted to utilize the method of boundary function and the thoery of differential inequalities to study the contrast structures of singular perturbation problems with integral boundary conditions.Firstly, we study the step-type contrast structure of second-order semilinear dif- ferential equation with integral boundary conditions: whereμis a small and positive parameter, and f:[0,1]×R→R, hi:R→R (i= 1,2) are C(2)-functions. The asymptotic solution is constructed by the boundary function method, and the uniform validity of the formal solution is proved by the extended theory of differential equalities.Secondly, we study the spik-type solution of second-order semilinear differential equation with integral boundary conditions: whereμis a small and positive parameter, and f:[0,1]×R→R, is sufficiently smooth, hi:R→R (i= 1,2) are C2-function. The uniformly valid asymptotic solution is obtained by the boundary function method.
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