Existence Of Solutions For The Fully Third Order Nonlinear Boundary Value Problems

In this paper,by using the method of upper and lower solutions,fixed point the-orem of completely continuous operators and the fixed point index theory in cones,we discuss the existence and uniqueness of solutions and the existence of positive solutions to the two-point boundary value problem of fully third-order nonlinear differential equation(?)(?)where f:R x R3 ?R is a continuous function.The main results of this paper are as follows:1.With the aid of the existence and uniqueness of solutions for corresponding third-order linear boundary value problem,we obtain the existence and uniqueness of fully third-order two-point boundary value problem by using the Leray-Schauder fixed point theorem of completely continuous operators under the linear growt.h conditions.2.Under the Nagumo condition,we obtain the existence of fully third-order two-point boundary value problem via the lower and upper solution method and a special truncating technique.At the same time,under the linear and Nagumo-type growth conditions,we obtain the existence of the nonnegative solutions by using the the method of upper and lower solutions.3.Under the conditions concerning the first,eigenvalue of the corresponding third-order linear boundary value problem,the results of the existence of positive solutions for fully third-order nonlinear boundary value problem are obtained in the case of superlinear and sublinear by choosing a suitable cone and applying the fixed-point,index theory in cone.Specially,in the case of super linear,we obtain the solutions under the Nagumo-type growth conditions.