  In a rectangular domain, a boundary-value problem is considered for a
mixed-type equation with a regularized Caputo-like counterpart of hyper-Bessel
differential operator and the double-order Hilfer fractional derivative. Using
the method of separation of variables, Laplace transform, a unique solvability
of the considered problem has been established. Moreover, we have found an
explicit solution of initial problems for differential equations with the
double-order Hilfer's derivative and regularized Caputo-like counterpart of the
hyper-Bessel differential operator with the non-zero starting point.
