The stability of the fractional Volterra integro‐differential equation by means of Ψ‐Hilfer operator revisited

Abstract

In this note, we have as main purpose to investigate the Ulam‐Hyers stability of a fractional Volterra integral equation through the Banach fixed point theorem and present an example on Ulam‐Hyers stability using operator theory α‐resolvent in order to elucidate the investigated result. Our results modify the Theorem 4 of Sousa and et. al. [Sousa JVC, Rodrigues FG, Oliveira EC. Stability of the fractional Volterra integro‐differential equation by means of Ψ‐Hilfer operator. Math Meth Appl Sci. 2019;42(9):3033‐3043.] and present a corrected proof with a modified approximation.

Resolvent Positive Operators and Positive Fractional Resolvent Families

“…Applying Theorem 15 to the operator A + B, we obtain that A + B ∈ C α + , which implies that A + B ∈ A α + by Theorem 17.…”

“…There are also literatures devoted to abstract semilinear fractional Cauchy problems, see, e.g., [17,18]. We refer to the survey paper [19] and the references therein for the basic theory of abstract fractional differential equations.…”