Existence of optimal pairs and solvability of non-autonomous fractional Sobolev-type integrodifferential equations

This article establishes a sufficient criteria for the existence of an optimal pair and a mild-solution of a non-autonomous fractional Sobolev-type integro-differential equation (FSIDE) with a generalized nonlocal condition. The fixed-point technique, tools from fractional calculus and the semigroup-theory played a valuable role in deriving the results. It is imperative to mention the distinguish attributes of this work that the developed results don't assume (i) that the generated semigroup is compact; (ii) that the linear operators -S(t) are bounded; and (iii) that the operator D is strongly continuous. We also provide an example to demonstrate the established results.