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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.

39B42 Matrix and operator functional equations
Full Text: DOI
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