Direct and inverse problem of Cauchy type for the fraction equation with the degenerationand non-loaded operator coefficient.

Abstract

A sufficient condition is proved for the solvability of a Cauchy-type problem for a degenerate fractional-order abstract equation. A distinctive feature of the work is the fact that the domain of the operator coefficient determination, which characterizes the degeneracy, is not assumed to be dense one. According to the operator coefficients of the equation they develop some operator function belonging to the space of linear bounded operators, and with its help a unique solvability of direct and inverse problems of Cauchy type is established. Let's note that the degenerate inverse problem is considered for the first time.