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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 Δ = [ a , b ] and on the infinite interval  I  = [ a , ∞ ). Our analysis is based on the application of the Banach fixed‐point theorem and the Gronwall inequality involving generalized Ψ‐fractional integral. At last, we performed out an application to elucidate the outcomes got.

On the Ulam‐Hyers stabilities of the solutions of Ψ‐Hilfer fractional differential equation with abstract Volterra operator