The Existence Of Solutions For Boundary Value Problems Of Fractional Diferential Equation In Banach Spaces

In this paper, with the help of the measure of noncompactness and the partial order theory, we use the monotone iterative technique of upper and lower solu-tions, the fixed point theorem of condensing mapping and the fixed point index theory in cones to discuss the existence of solutions for boundary value problems of nonlinear fractional differential equation in Banach spaces where1<β≤2is real number, J=[0,1], D0-∞is the standard Riemann-Liouville fractional derivative,f:J×Eâ†'E is continuous,θ is the zero element of EThe main results of this paper are as follows:1. With the help of the existence of solutions for fractional differential equation in real number space, we get the corresponding existence and uniqueness of the fractional linear differential equation in Banach spaces.2. Based on the existence and uniqueness of solutions for fractional linear differ-ential equation in Banach spaces and the monotone iterative technique of upper and lower solutions, we obtain the corresponding results of the existence and uniqueness of solution for fractional nonlinear differential equation boundary value problems.3. Combining with the Sadovaskii fixed point theorem and Leray-Schauder type fixed point theorem of condensing mapping, we obtain the existence of solutions for fractional nonlinear differential equation boundary value problems under the condition of the measure of noncompactness in Banach spaces.4. By applying the fixed-point index theory of condensing mapping in cones, the existence results of positive solutions for fractional diferential equation boundaryvalue problems are obtained in order Banach spaces.