Implicit Fractional Differential Equation with Nonlocal Fractional Integral Conditions

In this paper, we study and investigate the following implicit Caputo fractional derivative and nonlocal fractional integral conditions of the from: D-C(0+)q u(t) = f (t, u(t), (c) D-0+(q) u(t)), t is an element of [0, T] u(0) = eta, u(T) = (RL) I-0+(p) u(kappa), kappa is an element of (0, T) where 1 < q <= 2, 0 < p <= 1, eta is an element of R, D-c(0+)q u(t) is the Caputo fractional derivative of order q, I-RL(0+)p is the Blemann-Liouville fractional integral of order p and f : [0, T] x R x R -> R is continuous function by using Krasnoselskii's fixed point theorem and Boyd-Wong non-linear contraction. Also, we study the existence and uniqueness of this problem. An example is established to support our main results.