The inverse problem with unknown source function is studied for the nonlinear Oskolkov's system of partial differential equations that describes the dynamics of viscoelastic Kelvin–Voigt fluid. It is reduced to the problem for a first order semilinear ordinary differential equation with operator coefficients in Banach spaces. The main difficulty of the problem is a presence of the operator with a nontrivial kernel at the derivative. By the results on the unique solvability of the abstract problem, obtained by the authors before, the existence of a unique classical solution for the Oskolkov's system inverse problem is proved. Copyright © 2015 John Wiley & Sons, Ltd.

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Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.