Generalized Solutions of Operator Equations and Extreme Elements

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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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.

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)

• , 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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