A qualitative study on generalized Caputo fractional integro-differential equations
Mohammed D. Kassim, Thabet Abdeljawad, Wasfı Shatanawi, Saeed M. Ali, Mohammed S. Abdo
Abstract
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 $(I_{a}^{\rho }f)(t)=\int _{a}^{t}f(s)s^{\rho -1}\,ds$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:msubsup><mml:mi>I</mml:mi><mml:mi>a</mml:mi><mml:mi>ρ</mml:mi></mml:msubsup><mml:mi>f</mml:mi><mml:mo>)</mml:mo><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:msubsup><mml:mo>∫</mml:mo><mml:mi>a</mml:mi><mml:mi>t</mml:mi></mml:msubsup><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>s</mml:mi><mml:mo>)</mml:mo><mml:msup><mml:mi>s</mml:mi><mml:mrow><mml:mi>ρ</mml:mi><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace/><mml:mi>d</mml:mi><mml:mi>s</mml:mi></mml:math> . 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.