We solve the Cauchy problem for inhomogeneous distributed-order equations in a Banach space with a linear bounded operator in the right-hand side, with respect to the distributed Caputo derivative.First we find the solution by using the unique solvability theorem for the Cauchy problem.Then the results obtained are applied to the analysis of a Others distributed-order system of ordinary differential equations.Then we study an analogous equation, but with degenerate linear operator at the distributed derivative, Flyers which is called a degenerate equation.
The pair of linear operators in the equation is assumed to be relatively bounded.For the two types of initial-value problems, we obtain the existence and uniqueness of a solution, and derive its form.Abstract results for the degenerate equations are used in the study of initial-boundary value problems with distributed order in time equations with polynomials of self-adjoint elliptic differential operator with respect to the spatial derivative.