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Concept of Generalized Solution of Nonlinear Operator Equation

In: Generalized Solutions of Operator Equations and Extreme Elements

Author

Listed:
  • 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)

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.

Suggested Citation

  • D. A. Klyushin & S. I. Lyashko & D. A. Nomirovskii & Yu. I. Petunin & V. V. Semenov, 2012. "Concept of Generalized Solution of Nonlinear Operator Equation," Springer Optimization and Its Applications, in: Generalized Solutions of Operator Equations and Extreme Elements, edition 1, chapter 0, pages 137-161, Springer.
    • Handle: RePEc:spr:spochp:978-1-4614-0619-8_7
    DOI: 10.1007/978-1-4614-0619-8_7
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