Nonlinear inverse problems for some equations with fractional derivatives

The solvability of nonlinear inverse problems with a time-dependent unknown element for evolution equations in Banach spaces with Gerasimov — Caputo derivatives is investigated. A theorem is obtained on the existence of a unique smooth solution of a nonlinear problem for an equation solved with respect to the highest fractional derivative with a bounded operator in the linear part. It is used in the study of degenerate evolution equations under the condition of $p$-boundedness of a pair of operators in the linear part of the equation — at the highest derivative and at the desired function. In the case of the action of a nonlinear operator into a subspace without degeneration, the existence of a unique smooth solution is proved; and for the independent of the nonlinear operator from elements of the degeneration subspace, the existence of a unique generalized solution is shown. The abstract results obtained for degenerate equations are used in the study of an inverse problem for a modified system of Sobolev equations with unknown coefficients at lower order fractional derivatives in time.