Abstract
<jats:p>In this paper, we study a nonlocal initial boundary value problem for an abstract time-fractional diffusion-wave equation involving an unbounded, positive, self-adjoint operator on a Hilbert space. The existence of a mild solution is investigated using the eigenfunction decomposition method. By expanding the solution in terms of the operators eigenfunctions, the problem is reduced to a system of ordinary fractional differential equations with nonlocal conditions. The solutions of these equations are expressed in terms of the Mittag-Leffler function. By examining the zeros of the denominator, we identify the appropriate interval of definition, which excludes the right endpoint, ensuring the correctness of the solution. The solution to the original problem is expressed as a series expansion in terms of the eigenfunctions of an abstract operator. By applying estimation techniques in the corresponding Hilbert spaces, we establish the existence, uniqueness, and regularity of the solution.</jats:p>