Solvability Of Boundary Value Problems For Third-Order Ordinary Differential Equations In Banach Spaces

In this paper,with the aid of the measure of noncompactness and the partial order theory,by using the fixed point,theorem of condensing mapping,we deal with the solvability of boundary value problem for the full third-order ordinary differential equation in Banach space E,where I =[0,1],f:I × E3 ? E is continuous,? is null element of E.Moreover,by using the monotone iterative technique of upper and lower solutions and the fixed point index theory of condensing mapping in cones,we discuss the solvability of boundary value problem for third-order ordinary differential equation in Banach space E,where 1 =[0,1],f:I × E3 ? E is continuous,? is null element of E.The main results of this paper are as follows:1.With the aid of the existence theorem of solutions of boundary value problem for the full linear third-order ordinary differential equation in R,we obtain the existence and uniqueness of solutions for corresponding boundary value problem of linear third-order ordinary differential equation in Banach space.2.Under the linear growth conditions,by using the Leray-Schauder fixed point theorem,Sadovskii fixed point theorem and Banach fixed point theorem of condens-ing mapping with the aid of measure of noncompactness,we obtain the existence and uniqueness of solutions for boundary value problem of the full third-order ordinary differential equation in Banach space.3.Under the measure of noncompactness conditions,by establishing the new maximum principle,combining with the monotone iterative technique of upper and lower solutions,we obtain the existence and uniqueness of solutions for boundary value problem of third-order ordinary differential equation in Banach space.4.Under the conditions that the nonlinear term f satisfies supcrlinear or sublin-ear growth,by constructing a special cone and applying the measure of noncompact-ness estimation techniques and the fixed point index theory of condensing mapping in cone,we obtain the existence of positive solutions for boundary value problem of third-order ordinary differential equation in Banach space.