Degenerate fractional differential equations in locally convex spaces with a $\sigma$-regular pair of operators

We consider a degenerate fractional order differential equation $D^\alpha_tLu(t)=Mu(t)$ in a Hausdorff secquentially complete locally convex space is considered. Under the $p$-regularity of the operator pair $(L,M)$, we find the phase space of the equation and the family of its resolving operators. We show that the identity image of the latter coincides with the phase space. We prove an unique solvability theorem and obtain the form of the solution to the Cauchy problem for the corresponding inhomogeneous equation. We give an example of application the obtained abstract results to studying the solvability of the initial boundary value problems for the partial differential equations involving entire functions on an unbounded operator in a Banach space, which is a specially constructed Frechét space. It allows us to consider, for instance, a periodic in a spatial variable $x$ problem for the equation with a shift along $x$ and with a fractional order derivative with respect to time $t$.