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Generalized Solutions of Operator Equations and Extreme Elements

  • Book
  • © 2012

Overview

Authors:
  1. D.A. Klyushin

    1. , Department of Cybernetics, Kyiv National Taras Shevchenko Universit, Kyiv, Ukraine

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  2. S.I. Lyashko

    1. University of Kiev, Dept. Cybernetics, National Taras Shevchenko, Kiev, Ukraine

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  3. D.A. Nomirovskii

    1. , Department of Cybernetics, Kyiv National Taras Shevchenko Universit, Kyiv, Ukraine

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  4. Yu.I. Petunin

    1. University of Kiev, Dept. Cybernetics, National Taras Shevchenko, Kiev, Ukraine

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  5. V.V. Semenov

    1. , Department of Cybernetics, Kyiv National Taras Shevchenko Universit, Kyiv, Ukraine

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  • Contains recently obtained results in linear operator theory with applications to optimal control theory
  • Provides new methods of identifying solutions of linear and nonlinear operator equations
  • Can serve as a text in graduate courses and seminars in functional analysis and optimal control theory
  • Includes supplementary material: sn.pub/extras

Part of the book series: Springer Optimization and Its Applications (SOIA, volume 55)

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About this book

Abstract models for many problems in science and engineering take the form of an operator equation. The resolution of these problems often requires determining the existence and uniqueness of solutions to these equations. "Generalized Solutions of Operator Equations and Extreme Elements" presents recently obtained results in the study of the generalized solutions of operator equations and extreme elements in linear topological spaces. The presented results offer new methods of identifying these solutions and studying their properties. These new methods involve the application of a priori estimations and a general topological approach to construct generalized solutions of linear and nonlinear operator equations. The monograph is intended for mathematicians, graduate students and researchers studying functional analysis, operator theory, and the theory of optimal control.

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xxi
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  2. The Major Definitions, Concepts and Auxiliary Facts

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 1-5

  3. The Simplest Schemes of Generalized Solution of Linear Operator Equation

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 7-15

  4. A Priori Estimates for Linear Continuous Operators

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 17-27

  5. Applications of the Theory of Generalized Solvability of Linear Equations

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 29-101

  6. Computation of Near-Solutions of Operator Equations

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 103-124

  7. General Scheme of the Construction of Generalized Solutions of Operator Equations

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 125-136

  8. Concept of Generalized Solution of Nonlinear Operator Equation

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 137-161

  9. Generalized Extreme Elements

    • D. A. Klyushin, S. I. Lyashko, D. A. Nomirovskii, Yu. I. Petunin, V. V. Semenov
    Pages 163-194

  10. Back Matter

    Pages 195-202
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Reviews

From the reviews:

“This monograph principally deals with a unified approach to the notion of generalized solutions for linear and nonlinear operator equations between Banach spaces X and Y and, to a smaller degree, to the notion of generalized solutions to extremal problems for real functions defined on a set M of a Banach space. … this book is written sufficiently well and is easy readable. Undoubtedly, it will be interesting for all specialists in Functional Analysis and Partial Differential Equations.” (P. P. Zabreĭko, Mathematical Reviews, Issue 2012 j)

“This is a well-written, self-contained and easily readable monograph, with a reasonable number of references to English written papers and books. It can be recommended to everyone interested in the topic and having some background in functional analysis and topology.” (Robert Plato, Zentralblatt MATH, Vol. 1240, 2012)

Authors and Affiliations

  • , Department of Cybernetics, Kyiv National Taras Shevchenko Universit, Kyiv, Ukraine

    D.A. Klyushin, D.A. Nomirovskii, V.V. Semenov

  • University of Kiev, Dept. Cybernetics, National Taras Shevchenko, Kiev, Ukraine

    S.I. Lyashko, Yu.I. Petunin

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