Applications Of Differential Inequality In Several Nonlinear Boundary Value Problems

In this paper, existence(or uniqueness) results of some classes of BVP(without small parametre) were achieved via the method of upper and lower solution firstly, furthermore, we apply some of the existence results(or uniquness) to singlar perturbed BVP which contain small parametre, these kinds of BVP arrising frequently in mathematics and physics. By this way, existence of solution for singlar perturbed BVP can be gained, and the asymptotic behavior of solution as ε→ 0 can be studied additionly.This paper is arranged followingIn section 1, the concept of lower solution , upper solution and Nagumo condition were showed, and some princial theorems (Lemma 1.4,1.5) of second-order differential inequality were introducted, this concept and lemmas cite from ([l][2][12]),the Lemmas will be utilized in the section 2.In section 2, we construct the higher-order asymptotic solution of two kinds of nonlinear quadratic singular perturbed BVP by the method of boundary layer function,then existence of solution of nonlinear quadratic singlar perturbed BVP and the estimate of asymptotic solution about exact solution were achieved directly via the application of Lemma 1.5 which was showed in section 1.In section 3, Firstly, existence and uniqueness of solution of third order nonlinear Robin BVP was showed, then existence and uniqueness of solution of third order nonlinear Robin BVP with a turning point was achieved via the application of the existence and uniquness results mentioned above.In section 4, existence of solution of a class of nonlinear three-point BVP for nonlinear third-order differential equation was given, the analysis was based on the method of upper and lower solution. As a corollary, existence of solution of a class of third-order BVP with periodic boundary condition was gained.In section 5 , existence of solution of a class of nonlinear second-order differential systems for three-point BVP was given, the analysis was based on the method of upper and lower solution. As a corollary, existence of solution of a class of three-point BVP for forth-order differential equation was show; As an application, considered a type of forth-order semilinear singular perturbed three point BVP.