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A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation. (English. Russian original) Zbl 1345.39013
Russ. Math. 56, No. 3, 55-65 (2012); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2012, No. 3, 62-73 (2012).
Summary: 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 andminorant equations. We give examples of the reduction of controlled initial boundary value problems to the equation under consideration.

MSC:
39B42 Matrix and operator functional equations
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