On the total preservation of univalent global solvability for a first kind operator equation with controlled added nonlinearity

For a Cauchy problem associated with an evolutionary operator equation of first kind with controlled additional term depending nonlinearly on a phase variable in a Banach space we obtain sufficient conditions of the total (with respect to a whole set of admissible controls) preservation of univalent global solvability, and also uniform estimate for solutions. As examples we consider initial boundary value problems associated with a pseudoparabolic equation and a system of Oskolkov equations.