A majorant criterion for the total preservation of global solvability of controlled functional operator equation

For a nonlinear controlled functional operator equation in the Banach ideal space we prove a uniqueness theorem and also a theorem concerning sufficient conditions of global solvability for all controls from a pointwise bounded set. The second of these two theorems is proved subject to global solvability of some majorant equation for the control family mentioned above. The procedure of reduction of controlled initial boundary value problems to the equation under study is illustrated by examples.