Inhomogeneous degenerate Sobolev type equations with delay

Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate operator at the derivative, a relatively $p$-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations.