Concept of Generalized Solution of Nonlinear Operator Equation

D. A. Klyushin (), S. I. Lyashko (), D. A. Nomirovskii (), Yu. I. Petunin () and V. V. Semenov ()
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D. A. Klyushin: Kyiv National Taras Shevchenko University
S. I. Lyashko: Kyiv National Taras Shevchenko University
D. A. Nomirovskii: Kyiv National Taras Shevchenko University
Yu. I. Petunin: Kyiv National Taras Shevchenko University
V. V. Semenov: Kyiv National Taras Shevchenko University

Chapter Chapter 7 in Generalized Solutions of Operator Equations and Extreme Elements, 2012, pp 137-161 from Springer

Abstract: Abstract The purpose of Chap. 7 is to give a natural definition of generalized solution of non-solvable non-linear equation. In Sect. 7.1 we give the definition of generalized solution of equation with operator which acts in metric spaces. In Sect. 7.2 we give the definition of new concept of near-solution of non-linear operator equation. In Sect. 7.3 and 7.4 the existence, uniqueness, and correctness of generalized solution are proved. In Sect. 7.5 and 7.6 we investigate issues of embedding of metric spaces and extension of operators which are important for the theory of generalized solutions. Sections 7.7 and 7.8 contain examples of operators and describe numerical aspects of computation of generalized solutions. Section 7.9 is devoted to the development of the theory of generalized solutions of equations in uniform spaces and proximity spaces.

Keywords: Generalize Solution; Operator Equation; Basic Space; Continuous Operator; Cauchy Sequence (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-1-4614-0619-8_7

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