Existence and uniqueness of solutions for fractional Volterra-Fredholm equations in Banach spaces of order $\eta \in (1,2)$

The primary objective of this paper is to investigate and establish existence and uniqueness results for solutions of nonlinear Volterra-Fredholm integro-differential equations (VFIDEs) of fractional order, specifically for $1<\eta <2$. By leveraging fixed-point theorems and contraction mapping principles within Banach spaces, we derive comprehensive results for both one-dimensional and two-dimensional nonlinear fractional-order equations. By presenting sufficient conditions, we ensure the existence and uniqueness of a fixed point associated with the operator form of the VFIDEs. Our analysis provides a rigorous framework for understanding the behavior of such equations, and the results obtained in this study enhance our knowledge of fractional integro-differential equations (FIDEs). To illustrate the practical application of these theoretical results, two examples are provided that demonstrate the uniqueness of solutions.