Solvability Of Periodic Boundary Value Problems For Fractional Differential Equations In Banach Spaces

As we all know,most of the existence results of fractional order differential equations are obtained in real space R,but are still fresh in the abstract Banach space.Moreover,we propose that the conditions of nonlinear term f are harsh and have not reached the optimal conditions similar to those in integer order differential equations.In view of thisusing new tools,methods and techniques,this paper dis-cusses the existence of solutions to periodic boundary value problems for fractional differential equations in Banach spaces Secondly.we obtain the existence of positive solutions for periodic boundary value problem of fractional differential equations under wilder growth conditions in Banach spaces.The main results of this paper are as follows:1.Under the condition that the nonlinear term satisfies the weak conditions,we obtain the existence and uniqueness of solutions for periodic boundary value problem of fractional differential equation,by using monotone iterative technique of upper and lower solutions.2.By usiing the fixed point theorem of condensed mappings and the estimation technique of noncompactness measure,we obtain the existence of solutions to pe-riodic boundary value problems of fractional differential equation under conditions that the nonlinear term satisfies the first order growth.3.Under broader growth conditions and a new estimation technique of non-compactness measure,the existence of positive solutions for periodic boundary val-ue problem of fractional differential equations is achieved by selecting special cones and combining the fixed point index theory of condensed mappings.Morevoer,we obtain the condition that the nonlinear term is relatively optimal.