<p>Nonlinear inverse problems for differential equations in Banach spaces, which are solved with respect to the highest-order Dzhrbashyan–Nersesyan fractional derivative, are considered. A linear closed operator in the equation under study generates analytic resolving families for the corresponding linear homogeneous equation, and a Lipschitz continuous nonlinear operator in the equation depends on lower-order Dzhrbashyan–Nersesyan derivatives and on a time-dependent unknown parameter. The equation is endowed by the Dzhrbashyan–Nersesyan initial value conditions and an overdetermination condition, which is set by a linear bounded operator. By means of the Banach theorem on a fixed point, the existence of a unique global solution, both mild and classical, of the inverse problem is proved. The obtained general results are used for the solvability research of a nonlinear inverse problem to an equation of anomalous diffusion.</p>