Existence And Uniqueness Of Solutions For BVP Of Nonlinear Fractional Differential Equation

Fractional calculus has wide applications in many fields of science and engineer-ing, for example, biosciences, rheology, chemical physics, control theory of dynamical systems, optics and signal processing and so on. Recently, nonlinear fractional dif-ferential equations have been discussed under many different boundary conditions(BCs for short), for example, integer derivative BCs, integer derivative and integral BCs, integer and fractional derivative BCs, integer derivative and fractional integral BCs and nonlinear BCs.In this paper, we study the following boundary value problem (BVP for short)of nonlinear fractional differential equation with fractional integral BCs as well as integer and fractional derivative (?) where 2 < q < 3, 0 < ?1? 2, ?2 > 0, CD0+q and CD0+?1 denote the standard Caputo fractional derivative, I0+?2 denotes the standard Riemann-Liouville fractional integral and f ? C([0,1] ×R, R). By using nonlinear alternative of Leray-Schauder type and Banach contraction principle, we obtain the existence and uniqueness of solutions for the above BVP when 0 < ?1? 1 and 1 < ?1? 2, respectively.