In this paper, we study the existence of solutions for a new class of boundary value problems of non-linear fractional integro-differential equations. The existence result is obtained with the aid of Schauder type fixed point theorem while the uniqueness of solution is established by means of contraction mapping principle. Then, we present some examples to illustrate our results.

In this paper, we investigate the existence and uniqueness of solution for a multi-term fractional integro-differential problem with nonlocal four-point fractional boundary conditions via the Caputo differentiation. We obtain operational matrix of Riemann-Liouville fractional integral operator of Bernstein polynomials and investigate the numerical solutions of the problem by using the collocation method. By appling these matrices fractional integrodifferential equations convert to a linear system of equations. In this way, we give some examples to illustrate our results. The numerical method is computer oriented and produces very accurate and stable numerical results.c D q x(0) = − c D q x(T ), H. Bazgir, B. Ghazanfari: Equal contributor.