In this paper, we consider the new class of the fractional differential equation involving the abstract Volterra operator in the Banach space and investigate existence, uniqueness and stabilities of Ulam--Hyers on the compact interval \(\Delta=[a,b]\) and on the infinite interval \(I=[a,\infty )\). Our analysis is based on the application of the Banach fixed point theorem and the Gronwall inequality involving generalized \(\Psi\)-fractional integral. At last, we performed out an application to elucidate the outcomes got.