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**Homotopy of extremal problems. Theory and applications.**
*(English)*
Zbl 1141.49001

de Gruyter Series in Nonlinear Analysis and Applications 11. Berlin: de Gruyter (ISBN 978-3-11-018942-1/hbk). xi, 303 p. (2007).

This monograph presents a literal translation of the same titled book in Russian that was published by “Nauka” in 2001. The monograph is devoted to the applications of the homotopy method (with the combination with method of a continuation with respect to parameters) in the different classes of extremal problems. The general scheme of applying the homotopy method in concrete problems involves so called deformation principle for a minimizer. This principle provides the invariance of a local minimizer for the family of functions with respect to some sorts of deformations and in this way it allows one to conclude about the local properties of the solutions of corresponding problems. The monograph contains applications of the above mentioned method to abstract finite- and infinite-dimensional problems, classical problems of the various calculus, multi-valued variational problems etc.

The book 5 has chapters. In its first chapter the authors expound preliminaries from different branches of topology, classical and modern functional analysis (including convex and non-smooth analysis), extremal problems theory. Chapter 2 acquaints the reader with the homotopy method using as example non-degenerate deformations of finite-dimensional functions. The authors prove several statements of the deformation theorem (deformation principle for a minimizer) for different function classes. Also converses of these theorems are discussed. In parallel with non-degenerate deformations authors also concern generalized non-degenerate deformations, i.e. one-parameter families of functions with a non-degenerate gradient on the boundary of the critical point neighborhood. To investigate the existence of such deformations, the authors employ the basics of the degree theory, including the Hopf and Parusinski theorems with the proofs. In the Chapter 3 the ideas of the homotopy method extended over infinite-dimensional problems. The authors prove the deformation theorems for some functional classes in Hilbert and Banach spaces, for special classes of the Lipschitzian functionals and for some non-smooth extremal problems with the restrictions.

In the Chapter 4 the basics of the homotopy Conley index are recounted. The correctness of all definitions and basic properties of the homotopy Conley index are stated and proved with the utmost clarity and accuracy. The proof of the homotopy invariance of Conley index is also adduced. To apply the Conley index theory to infinite-dimensional problems the authors introduce the special class of functionals, so called \((E,H)\)-regular functionals. This class of functionals was firstly described by M. A. Krasnosel’ski and N. A. Bobylev and proved to be useful for the investigating extremals in multidimensional variational problems. For the critical points of \((E,H)\)-regular functionals, the homotopy invariance of Conley index is also established.

Finally, the largest Chapter 5 of the book is devoted to the applications of the homotopy method and the theory of Conley index in various types of extremal problems. Among discussed applications there are the ones to the problems of the nonlinear programming and calculus of variations, problems of the optimal control theory and some multicriteria problems. Also by means of the homotopy method the authors investigate the solution stability of ODE’s and the bifurcations of extremals in variational problems.

It is worth to remark, that the exploring this monograph does not demand from the reader any specific and in-depth knowledge of the nonlinear analysis: all the definitions and theorems referred in the book are given and thoroughly explained in the main text. According to this, the book is quite comprehensive not only for the specialists in the branches of the nonlinear analysis, so it could be considered as the textbook for the students and post-graduates in mathematics.

The book 5 has chapters. In its first chapter the authors expound preliminaries from different branches of topology, classical and modern functional analysis (including convex and non-smooth analysis), extremal problems theory. Chapter 2 acquaints the reader with the homotopy method using as example non-degenerate deformations of finite-dimensional functions. The authors prove several statements of the deformation theorem (deformation principle for a minimizer) for different function classes. Also converses of these theorems are discussed. In parallel with non-degenerate deformations authors also concern generalized non-degenerate deformations, i.e. one-parameter families of functions with a non-degenerate gradient on the boundary of the critical point neighborhood. To investigate the existence of such deformations, the authors employ the basics of the degree theory, including the Hopf and Parusinski theorems with the proofs. In the Chapter 3 the ideas of the homotopy method extended over infinite-dimensional problems. The authors prove the deformation theorems for some functional classes in Hilbert and Banach spaces, for special classes of the Lipschitzian functionals and for some non-smooth extremal problems with the restrictions.

In the Chapter 4 the basics of the homotopy Conley index are recounted. The correctness of all definitions and basic properties of the homotopy Conley index are stated and proved with the utmost clarity and accuracy. The proof of the homotopy invariance of Conley index is also adduced. To apply the Conley index theory to infinite-dimensional problems the authors introduce the special class of functionals, so called \((E,H)\)-regular functionals. This class of functionals was firstly described by M. A. Krasnosel’ski and N. A. Bobylev and proved to be useful for the investigating extremals in multidimensional variational problems. For the critical points of \((E,H)\)-regular functionals, the homotopy invariance of Conley index is also established.

Finally, the largest Chapter 5 of the book is devoted to the applications of the homotopy method and the theory of Conley index in various types of extremal problems. Among discussed applications there are the ones to the problems of the nonlinear programming and calculus of variations, problems of the optimal control theory and some multicriteria problems. Also by means of the homotopy method the authors investigate the solution stability of ODE’s and the bifurcations of extremals in variational problems.

It is worth to remark, that the exploring this monograph does not demand from the reader any specific and in-depth knowledge of the nonlinear analysis: all the definitions and theorems referred in the book are given and thoroughly explained in the main text. According to this, the book is quite comprehensive not only for the specialists in the branches of the nonlinear analysis, so it could be considered as the textbook for the students and post-graduates in mathematics.

Reviewer: Peter Zabreiko (Minsk)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

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