Abstract
In this paper, we consider one family of problems simulating the determination of target components and density of sources from given values of the initial and final states. The mathematical statement of these problems leads to the inverse problem for the diffusion equation, where it is required to find not only a solution of the problem, but also its right-hand side that depends only on a spatial variable. One of specific features of the considered problems is that the system of eigenfunctions of the multiple differentiation operator subject to boundary conditions of the initial problem does not have the basis property. The other specific feature of the considered problems is that an unknown function is simultaneously present both in the right-hand side of the equation and in conditions of the initial and final redefinition. We prove the unique existence of a generalized solution to the mentioned problem.