Qualitative Analysis of Nonlinear Generalized Caputo Fractional Volterra-Fredholm System - Issa, Mohammed S. Bani (Discontinuity, Nonlinearity, and Complexity, 2026)
The nonlocal Volterra-Fredholm system for a class of nonlinear fractional $ψ$-Caputo integro-differential equations of Sobolev type in Banach spaces is examined in this study. The $ψ$-Caputo fractional derivative generalizes the classical Caputo derivative by incorporating a function $ψ$, allowing greater flexibility in modeling memory effects. We first establish novel Darbo-type fixed point theorems. These theorems are then applied to derive a solvability theorem for the nonlocal Volterra-Fredholm problem associated with nonlinear fractional equations of Sobolev type. Furthermore, we provide an example to show how our theoretical findings might be applied. These results have potential applications in physics and engineering, particularly in modeling complex systems with memory and hereditary properties.