@article{Jeelani2025, 
author = {Mdi Begum Jeelani and Farva Hafeez and Nouf AbdulRahman Alqahtani},
title = {Existence and uniqueness of solutions for fractional Volterra-Fredholm equations in Banach spaces of order    η  ∈  (  1  ,  2  )},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {9},
pages = {21916-21928},
keywords = {fixed point theorem, Caputo derivative, Volterra-Fredholm integro-differential equations, Riemann-Liouville derivative},
url = {https://www.sciopen.com/article/10.3934/math.2025976},
doi = {10.3934/math.2025976},
abstract = {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  <  η  <  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.}
}