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A qualitative study on generalized Caputo fractional integro-differential equations

Authors
  • Kassim, Mohammed D.1
  • Abdeljawad, Thabet2, 3, 4
  • Shatanawi, Wasfi2
  • Ali, Saeed M.1
  • Abdo, Mohammed S.5
  • 1 Imam Abdulrahman Bin Faisal University, P.O. Box 1982, Dammam, 34151, Saudi Arabia , Dammam (Saudi Arabia)
  • 2 Prince Sultan University, Riyadh, Saudi Arabia , Riyadh (Saudi Arabia)
  • 3 China Medical University, Taichung, 40402, Taiwan , Taichung (Taiwan)
  • 4 Asia University, Taichung, Taiwan , Taichung (Taiwan)
  • 5 Hodeidah University, Al-Hudeidah, Yemen , Al-Hudeidah (Yemen)
Type
Published Article
Journal
Advances in Difference Equations
Publisher
Springer International Publishing
Publication Date
Aug 10, 2021
Volume
2021
Issue
1
Identifiers
DOI: 10.1186/s13662-021-03530-6
Source
Springer Nature
Keywords
Disciplines
  • Fixed Point Theory and Applications to Fractional Ordinary and Partial Difference and Differential E
License
Green

Abstract

The aim of this article is to discuss the uniqueness and Ulam–Hyers stability of solutions for a nonlinear fractional integro-differential equation involving a generalized Caputo fractional operator. The used fractional operator is generated by iterating a local integral of the form (Iaρf)(t)=∫atf(s)sρ−1ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$\end{document}. Our reported results are obtained in the Banach space of absolutely continuous functions that rely on Babenko’s technique and Banach’s fixed point theorem. Besides, our main findings are illustrated by some examples.

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