Abstract: In this paper,the Nagumo theorem and the fixed-point theorem are used to prove the exist-  ence and the uniqueness and to estimate the asyrmptotic expansion of the shock solutions of the boundary value problems for a class of quasilinear differential equations,the asymptotic expansion of solution of  any orders including boundary is obtained.

Key words: quasilinear , equation, Nagumo , theorem, fixed-point , theorem, asymptotic ,  , solution