Existence of ‎W‎eak Solutions to a Kind of System of Fractional Semi-Linear Fredholm-Volterra Boundary Value Problem

This article is devoted to study the weak solutions of a class of nonlinear system of fractional boundary value problems including both volterra and fredholm linear integral terms. this system of fractional semi-linear fredholm-volterra integro-differential equations does have a gradient of a nonlinear source term as well. we apply the critical point theory and the variational structure to prove the existence of at least three distinct weak solutions to the system. furthermore, it is presented an example to verify the legitimacy and applicability of the theory.