The nonlinear \(p\) -Laplacian operator appears in a wide range of real-world problems, from mechanics to image analysis and game theory. In this work, we focus on a new type of fractional integro-differential equation that brings together the \(g\) -Caputo generalized proportional derivative and the \(p\) -Laplacian, forming a novel mathematical framework. This combination makes it possible to capture both memory effects and nonlinear behaviors in dynamic systems. To study the problem, we apply key results from fractional calculus and use fixed-point techniques specifically, Mönch’s and Banach’s theorems to prove the existence and uniqueness of solutions. A numerical example is included to demonstrate the applicability of the theoretical findings. The approach presented here opens new possibilities for modeling phenomena in fields such as anomalous diffusion, viscoelastic materials, and nonlinear control systems.
