A note on Sobolev form fractional integro-differential equation with state-dependent delay via resolvent operators

Abstract

This paper explores the  new existence and uniqueness of mild solutions for a class of Sobolev form  fractional  integro-differential equation (in short SFFIDE) with state-dependent  delay (in short SDD)  and   nonlocal conditions (in short NLCs) via resolvent  operators in Banach spaces. By making use of  Banach contraction principle and Krasnoselskii fixed point theorem (in short FPT) along with resolvent operators and fractional calculus, we develop the sought outcomes. To obtain our results, our working hypotheses are that the functions determining the equation satisfy certain Lipschitz conditions of local type. An illustration  is furthermore provided to demonstrate the acquired concepts.