Specific Cases of One General Inverse Problem for Abstract Differential Equations.

We consider a general statement of linear inverse problem for -th order abstract differential equation in a Banach space. A universal uniqueness criterion for a solution of this problem is given. It is expressed through zeros of special Mittag-Leffler functions. The main emphasis is placed on the cases when all zeros of the functions are found explicitly and the uniqueness criterion takes an elementary form. It is shown that the previous results for equations of the st and nd orders follow from the general criterion. Fundamentally new is the case of one inverse problem for the th order differential equation, where the result also takes an elementary form. This last case is discussed in detail in our paper. [ABSTRACT FROM AUTHOR]