On the Relationship between the Solutions of an Abstract Euler-Poisson-Darboux Equation and Fractional Powers of the Operator Coefficient in the Equation

We consider an incomplete initial value problem for an abstract singular Euler-Poisson-Darboux equation. It is established that, under weakened requirements on the operator coefficient in the equation, fractional powers of this operator coefficient should be used to construct solutions. It is also shown that a fractional power of the operator coefficient relates the Dirichlet condition and the weighted Neumann condition in the case of a boundary value problem for the Euler-Poisson-Darboux equation.