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**Generalized inverse operators and Fredholm boundary-value problems. Translated from the Russian by P. V. Malyshev and D. V. Malyshev.**
*(English)*
Zbl 1083.47003

Utrecht: VSP (ISBN 90-6764-407-2/hbk). xiv, 317 p. (2004).

This monograph is devoted to generalized inverse operators and Fredholm boundary value problems. For this type of boundary value problems, the authors give a natural classification of critical and noncritical cases, establish sufficient conditions for the coefficients guaranteeing the existence of solutions, and develop iterative algorithms for the construction of solutions of these problems. The book is divided into 7 chapters.

In Chapter 1 (Preliminary Information), some well-known definitions and results from functional analysis, the theory of linear operators in Hilbert and Banach spaces and matrix theory which are required for subsequent chapters are presented. In Chapter 2 (Generalized Inverse Operators in Banach Spaces), the authors study the problems encountered in the construction of generalized inverse operators for linear bounded Fredholm operators acting in Banach spaces. This construction is realized by complementing an original operator \(L\) to an operator \(\overline L\) of complete rank. In Chapter 3 (Psevdoinverse Operators in Hilbert Spaces), methods for the construction of generalized inverse operators in Hilbert spaces are studied. In Chapter 4 (Boundary-Value Problems for Operator Equations), the theory of generalized inversion and pseudoinversion of linear Fredholm operators in Banach and Hilbert spaces developed in the previous chapters enable the authors to develop a unified procedure for the investigation of Fredholm boundary-value problems for operator equations solvable either everywhere or not everywhere. The proposed approach is then improved for the analysis of boundary-value problems for standard operator systems, including systems of ordinary differential equations and equations with delay in Chapter 5 (Boundary Value Problems for Ordinary Differential Equations) and systems with impulsive action in Chapter 6 (Impulsive Boundary Value Problems for Systems of Ordinary Differential Equations). Finally, in Chapter 7 (Solutions of Differential and Difference Systems Bounded on the Entire Real Axis), the authors obtain necessary and sufficient conditions for the existence of solutions of linear and nonlinear differential and difference systems bounded on the entire axis. The bibliography contains 158 entries.

In Chapter 1 (Preliminary Information), some well-known definitions and results from functional analysis, the theory of linear operators in Hilbert and Banach spaces and matrix theory which are required for subsequent chapters are presented. In Chapter 2 (Generalized Inverse Operators in Banach Spaces), the authors study the problems encountered in the construction of generalized inverse operators for linear bounded Fredholm operators acting in Banach spaces. This construction is realized by complementing an original operator \(L\) to an operator \(\overline L\) of complete rank. In Chapter 3 (Psevdoinverse Operators in Hilbert Spaces), methods for the construction of generalized inverse operators in Hilbert spaces are studied. In Chapter 4 (Boundary-Value Problems for Operator Equations), the theory of generalized inversion and pseudoinversion of linear Fredholm operators in Banach and Hilbert spaces developed in the previous chapters enable the authors to develop a unified procedure for the investigation of Fredholm boundary-value problems for operator equations solvable either everywhere or not everywhere. The proposed approach is then improved for the analysis of boundary-value problems for standard operator systems, including systems of ordinary differential equations and equations with delay in Chapter 5 (Boundary Value Problems for Ordinary Differential Equations) and systems with impulsive action in Chapter 6 (Impulsive Boundary Value Problems for Systems of Ordinary Differential Equations). Finally, in Chapter 7 (Solutions of Differential and Difference Systems Bounded on the Entire Real Axis), the authors obtain necessary and sufficient conditions for the existence of solutions of linear and nonlinear differential and difference systems bounded on the entire axis. The bibliography contains 158 entries.

Reviewer: Sotiris K. Ntouyas (Ioannina)

### MSC:

47A05 | General (adjoints, conjugates, products, inverses, domains, ranges, etc.) |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

34B15 | Nonlinear boundary value problems for ordinary differential equations |

34B37 | Boundary value problems with impulses for ordinary differential equations |

34K10 | Boundary value problems for functional-differential equations |

47A50 | Equations and inequalities involving linear operators, with vector unknowns |

47A53 | (Semi-) Fredholm operators; index theories |

47N20 | Applications of operator theory to differential and integral equations |