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

Abstract

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