Comprehensive analysis on the existence and uniqueness of solutions for fractional q-integro-differential equations.

In this work, we study the coupled system of fractional integro-differential equations, which includes the fractional derivatives of the Riemann–Liouville type and the fractional q-integral of the Riemann–Liouville type. We focus on the utilization of two significant fixed-point theorems, namely the Schauder fixed theorem and the Banach contraction principle. These mathematical tools play a crucial role in investigating the existence and uniqueness of a solution for a coupled system of fractional q-integro-differential equations. Our analysis specifically incorporates the fractional derivative and integral of the Riemann–Liouville type. To illustrate the implications of our findings, we present two examples that demonstrate the practical applications of our results. These examples serve as tangible scenarios where the aforementioned theorems can effectively address real-world problems and elucidate the underlying mathematical principles. By leveraging the power of the Schauder fixed theorem and the Banach contraction principle, our work contributes to a deeper understanding of the solutions to coupled systems of fractional q-integro-differential equations. Furthermore, it highlights the potential practical significance of these mathematical tools in various fields where such equations arise, offering a valuable framework for addressing complex problems. [ABSTRACT FROM AUTHOR]