Ulam–Hyers–Mittag-Leffler stability for a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si10.svg"> <mml:mi>ψ</mml:mi> </mml:math> -Hilfer problem with fractional order and infinite delay

Abstract

The research article concerned with developing the qualitative theory for nonlinear fractional functional differential equations (FFDEs) of arbitrary order with infinite delay involving generalized Hilfer fractional derivative of the form: HD0+α,β;ψy(t)=f(t,yt);t∈(0,b],I0+1−γ;ψy(0+)=y0,y(t)=φ(t);t∈(−∞,0].Here 0<α<1, 0≤β≤1, y0∈R, and D0+α,β;ψ, I0+1−γ;ψ are generalized fractional operators in the concepts Hilfer and Riemann–Liouville, respectively. Some new and recent results of existence and Ulam–Hyers–Mittag-Leffler (UHML) stability of solution for the proposed problem will also be highlighted. The concerned analysis is carried out via using the Banach fixed point theorem, Picard operator method, and generalized Gronwall’s inequality. Finally, an example is given to illustrate the effectiveness of our main results.