The Existence Of Solutions For A Class Of Boundary Value Problems Of Fractional Differential Equation In Banach Spaces

In this paper, we use the monotone iterative technique of upper and lower solutions, the fixed point theorem of condensing mapping and the fixed point index theory in cones to discuss the existence of solutions for a class of boundary value problems of nonlinear fractional differential equation in Banach spaces {-D0?+u(t)?f(t,u(t)),t?I,u(0)= u'(0)= u'(1)=?, where 2< ?? 3 is a real number,I= [0,1], D0+?is the standard Riemann-Liouville fractional derivative,f:I x E? E is continuous,? is the zero element of E.The main results of this paper are as follows:1. Firstly, we discussed the corresponding existence and uniqueness of solutions of boundary value problems of the linear fractional differential, and we obtained the norm estimate of the operator of solutions through careful calculations.2. Based on the monotone iterative technique of upper and lower solutions, we obtained the corresponding results of the existence and uniqueness of solution for nonlinear differential equation boundary value problems.3. Combining with the Sadovaskill fixed point theorem and Leray-Schauder type fixed point theorem of condensing mapping, we obtained the existence of solutions for nonlinear differential equation boundary value problems under the condition of the measure of noncompactness.4. By applying the fixed-point index theory of condensing mapping in cones, the existence results of positive solutions for nonlinear differential equation boundary value problems are obtained in order Banach spaces. Our results in this chapter extend and improve some known results. Evev in real number space is the best.