Several Classes Of Boundary Value Problems For Second-Order Quasilinear Differential Equations With Discontinuous Coefficients

In this paper we investigate several classes of singularly perturbed boundary value problems of second order quasilinear differential equations with discontinuous coef-ficients.By the theorem of lower and upper solutions we obtain the existence and asymptotic estimates of solution with interior layer for the proposed problem.First we consider the boundary value problems for second-order quasilinear differ-ential equations with discontinuous coefficients?u" + f(x,u)u' = g(x,u),0<x<1,u(0)= A,u(1)= B,where e is a small and positive parameter,A and B being given constants,and The functions f1,f2,g1,g2 are smooth enough on the interested domain,and f1(a,u)?f2(a,u).By the boundary function method and the theorem of upper and lower solu-tions we obtain the existence and asymptotic estimates of C1-smooth solution.Second,we study the following singularly perturbed boundary value problems of second order quasilinear differential equation?u"+f(x,u)u',g(x,u),0<x<1,u(0)= A,u(1)= B,whereHere f1(a,u)? f2(a,u)and g1(a,u)? g2(a,u),A and B are given constants.The left problem is second order quasilinear boundary value problem,and the right problem is semilinear Drichlet problem.By the boundary function method and the theorem of upper and lower solutions we obtain the existence and asymptotic estimates of C1-smooth solution.