Sobolev type fractional abstract evolution equations with nonlocal conditions and optimal multi-controls

This paper investigates the existence and uniqueness of mild solutions for a class of Sobolev type fractional nonlocal abstract evolution equations in Banach spaces. We use fractional calculus, semigroup theory, a singular version of Gronwall inequality and Leray–Schauder fixed point theorem for the main results. A new kind of Sobolev type appears in terms of two linear operators is introduced. To extend previous works in the field, an existence result of optimal multi-control pairs governed by the presented system is proved. Finally, an example is also given to illustrate the obtained theory.