On the Solvability of Initial and Boundary Value Problems for Abstract Functional-Differential Euler–Poisson–Darboux Equations

In a Banach space, we consider the Cauchy problem and the Dirichlet and Neumannboundary value problems for a functional-differential equation generalizing theEuler–Poisson–Darboux equation. A sufficient condition for the solvability of the Cauchy problemis proved, and an explicit form of the resolving operator is indicated, which is written using theBessel and Struve operator functions introduced by the author. For boundary value problems inthe hyperbolic case, we establish conditions imposed on the operator coefficient of the equationand the boundary elements that are sufficient for the unique solvability of these problems.