Complex powers of multivalued linear operators with polynomially bounded $C$-resolvent

TL;DR

This paper develops a theory for complex powers of multivalued linear operators with polynomially bounded $C$-resolvent, extending existing frameworks to non-injective $C$ and analyzing associated fractional differential inclusions in general locally convex spaces.

Contribution

It introduces a novel construction of complex powers for multivalued operators with polynomially bounded $C$-resolvent, applicable even when $C$ is not injective, and studies related fractional differential inclusions.

Findings

Constructed complex powers of multivalued operators with polynomially bounded $C$-resolvent.

Analyzed properties of these powers and their applications to fractional differential inclusions.

Extended the theory to general locally convex spaces, even in the Banach space setting.

Abstract

In the paper under review, we construct complex powers of multivalued linear operators with polynomially bounded $C$-resolvent existing on an appropriate region of the complex plane containing the interval $(-\infty,0].$ In our approach, the operator $C$ is not necessarily injective. We clarify the basic properties of introduced powers and analyze the abstract incomplete fractional differential inclusions associated with the use of modified Liuoville right-sided derivatives. We also consider abstract incomplete differential inclusions of second order, working in the general setting of sequentially complete locally convex spaces. Our results seem to be completely new even in the Banach space setting.