Stability of the fractional Volterra integro-differential equation by means of $ψ-$Hilfer operator

Published 7 Apr 2018 in math.CA | (1804.02601v1)

Abstract: In this paper, using the Riemann-Liouville fractional integral with respect to another function and the $\psi-$Hilfer fractional derivative, we propose a fractional Volterra integral equation and the fractional Volterra integro-differential equation. In this sense, for this new fractional Volterra integro-differential equation, we study the Ulam-Hyers stability and, also, the fractional Volterra integral equation in the Banach space, by means of the Banach fixed-point theorem. As an application, we present the Ulam-Hyers stability using the $\alpha$-resolvent operator in the Sobolev space $W^{{1,1}(\mathbb{R}_{+},\mathbb{C})$.}