Existence and Uniqueness of Nonlinear Volterra Integral Equations With Variable Fractional Order in Fréchet Spaces via a Frigon−Granas Fixed Point Approach

This paper investigates the existence and uniqueness of solutions to nonlinear Volterra integral equations of variable fractional order in Fréchet spaces. The variable-order fractional derivative is considered in the Riemann–Liouville sense, which extends classical approaches and is central to the paper’s novelty. By employing a nonlinear alternative of the Frigon–Granas fixed-point theorem for contraction mappings, a rigorous mathematical framework is provided, suitable for problems on semi-infinite intervals and for functions with variable fractional order, where classical Banach space approaches may fail. Illustrative examples demonstrate the applicability of the main results. The approach highlights the flexibility and generality of the method, paving the way for future extensions such as stability analysis and numerical schemes based on the established theory.