  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.
