Equations in Banach spaces with a degenerate operator under a fractional derivative

Abstract

We study an inhomogeneous linear fractional differential equation in a Banach space with a degenerate operator under the derivative. In the case of relative p-boundedness of the pair of operators in the equation, we prove the unique solvability of the Cauchy and Showalter problems for it and represent the solution via operator functions of the Mittag-Leffler type. The abstract results obtained are used to study a class of initial–boundary value problems for partial differential equations of arbitrary fractional order with respect to time.