A Cauchy Type Problem for a Degenerate Equation with the Riemann–Liouville Derivative in the Sectorial Case

Abstract

Under study is the unique solvability of a Cauchy type problem and a generalized Schowalter–Sidorov type problem for a class of linear inhomogeneous equations in Banach spaces with a degenerate operator at the Riemann–Liouville fractional derivative. We find an explicit form of a solution under some conditions for the pair of operators in the equation. To this end, we study a Cauchy type problem for an equation solvable with respect to the Riemann–Liouville derivative with an operator on the right-hand side which generates a resolving family of operators analytic in a sector. These abstract results are used to prove the unique solvability of an initial-boundary value problem for the Navier–Stokes system of equations of fractional order in time.