Existence of Mild Solution of the Hilfer Fractional Differential Equations with Infinite Delay on an Infinite Interval

In this study, we present a mild solution to the Hilfer fractional differential equations with infinite delay. Firstly, we establish the results on an infinite interval; to achieve this, we use the generalized Ascoli-Arzela theorem and Monch's fixed point theorem via a measure of noncompactness. Secondly, we consider the existence of a mild solution when the semigroup is compact, and the Schauder fixed-point theorem is used. The outcome is demonstrated using an infinitesimal operator, fractional calculus, semigroup theory, and abstract space. Finally, we present an example to support the results.