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

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Abstract

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.

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Authors and Affiliations

  1. Nizhni Novgorod State University, pr. Gagarina 23, Nizhni Novgorod, 603950, Russia

    A. V. Chernov

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  1. A. V. Chernov
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Correspondence to A. V. Chernov.

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Original Russian Text © A.V. Chernov, 2012, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2012, No. 3, pp. 62–73.

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Chernov, A.V. A majorant-minorant criterion for the total preservation of global solvability of a functional operator equation. Russ Math. 56, 55–65 (2012). https://doi.org/10.3103/S1066369X12030085

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