On the solution of a nonlinear fractional integro-differential equation with non-local boundary condition

This work studies the existence and the uniqueness of the solution to a kind of high-order nonlinear fractional integro-differential equations involving Rieman-Liouville fractional derivative. The boundary condition is of integral type which entangles ending point of the domain. First, the unique exact solution is extracted in terms of Green's function for the linear fractional differential equation and then Banach contraction mapping theorem is applied to prove the main result in the case of general nonlinear source term. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability.