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Nonlinear inverse problem for a general Sobolev type equation with depending on time overdetermination is researched in Banach spaces by means of the degenerate operator semigroups theory methods.The overdetermination condition is set on the solution semigroup image.Mild and strict solutions are proved to be existing and unique with Cauchy and generalized Showalter conditions on initial values.The abstract results allow to investigate nonlinear inverse evolution problems for Sobolev system of equations.