Fractional power of resolvent operators and applications in fractional neutral integro-differential equations with delay.

In this talk, we discuss recent advances in the theory of resolvent operators for abstract fractional differential equations. Our focus is on the construction and analysis of resolvent operators associated with abstract evolutionary fractional integro–differential equations involving the Caputo derivative of order $\alpha \in (0,1)$. Using tools from perturbation theory, we establish sufficient conditions for the existence and well-posedness of the corresponding fractional resolvent operators in Banach spaces.

By employing the concept of the fractional potential of a strongly continuous semigroup of linear operators, we derive fractional power estimates for the associated fractional resolvent operator. These estimates play a crucial role in the qualitative analysis of solutions and allow us to relax spatial regularity assumptions on the forcing terms appearing in the equations. As an application of these results, we prove the existence of mild solutions for a class of fractional neutral integro–differential equations with infinite delay, formulated in suitable phase spaces.

Finally, we illustrate the applicability of the developed theoretical framework by studying a coupled fractional partial integro–differential system involving the Caputo derivative. For this system, we establish the existence of mild solutions by combining the obtained resolvent estimates with fixed point arguments. The results presented in this talk provide a unified and robust approach to the analysis of fractional evolution equations with memory and delay effects, contributing to the ongoing development of the theory of fractional dynamical systems.