A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation

We study a nonlinear controlled functional operator equation in an ideal Banach space. We establish sufficient conditions for the global solvability for all controls from a given set, and obtain a pointwise estimate for solutions. Using upper and lower estimates of the functional component in the right-hand side of the initial equation (with a fixed operator component), we obtain majorant and minorant equations.We prove the stated theorem, assuming the monotonicity of the operator component in the right-hand side and the global solvability of both majorant and minorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.