To study existence of at least three weak solutions to a system of over-determined Fredholm fractional integro-differential equations

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In this article, the weak solutions of a class of nonlinear system of fractional boundary value problems including integral equations is investigated. Each equation in the system is either a nonlinear fractional partial integro-differential equation containing a gradient of a nonlinear source term or a Fredholm integral equation of the second kind. It is proved that there exist at least three distinct weak solutions by applying the critical point theory and the variational structure. In order to achieve this purpose, we use the well-known theorem on the construction of the critical set of functionals with a weak compactness condition. Furthermore, our main result is demonstrated by an illustrative example to show its legitimacy and applicability.