Lectures on nonlinear analysis.

*(English)*Zbl 1078.47001
Plzeň: Vydavatelský Servis (ISBN 80-86843-00-9). 353 p. (2004).

Nonlinear analysis is usually divided into topological methods, monotonicity methods, and variational methods. This natural methodological distinction is reflected by the “bible” of nonlinear analysis, namely E. Zeidler’s four volumes “Nonlinear functional analysis and its applications” published by Springer between 1985 and 1990 [Vol. I: “Fixed-point theorems” (1986; Zbl 0583.47050); Vol. II/A: “Linear monotone operators” (1990; Zbl 0684.47028); Vol. II/B: “Nonlinear monotone operators” (1990; Zbl 0684.47029); Vol. III: “Variational methods and optimization” (1985; Zbl 0583.47051); Vol. IV: “Applications to mathematical physics” (1988; Zbl 0648.47036)]. Together with the Russian book “Geometrical methods of nonlinear analysis” by M. A. Krasnosel’kij and P. P. Zabrejko [(Nauka, Moscow) (1975; Zbl 0326.47052); Engl. translation: (Springer, Berlin) (1984; Zbl 0546.47030)], and K. Deimling’s well-known monograph “Nonlinear functional analysis” [(Springer, Berlin) (1985; Zbl 0559.47040)], this was the standard reference for all specialists in nonlinear analysis up to 1990.

In the past five years, however, an increasing interest in nonlinear problems led to the publication of other monographs, devoted in part to special theoretical aspects, in part to more application-oriented topics in the field. Without aiming for a complete coverage, of course, we mention here “Elements of nonlinear analysis” by M. Chipot [(Birkhäuser, Basel) (2000; Zbl 0964.35002)], “Geometric nonlinear functional analysis” by Y. Benyamini and J. Lindenstrauss [(AMS, Providence/RI) (2000; Zbl 0946.46002)], “Nonlinear functional analysis” by W. Takahashi [(Yokohama Publ.) (2000; Zbl 0997.47002)], “Spectral theory and nonlinear functional analysis” by J. López–Gómez [(Chapman & Hall, Boca Raton/FL) (2001; Zbl 0978.47048)], “An introduction to nonlinear analysis” by Z. Denkowski, S. Migórski and N. S. Papageorgiou [(Kluwer, Dordrecht) (2003; Zbl 1040.46001)], “Nichtlineare Funktionalanalysis. Eine Einführung” by M. Růžička [(Springer, Berlin) (2004; Zbl 1043.46002)], “A topological introduction to nonlinear analysis” by R. F. Brown [(Birkhäuser, Boston/MA) (\({}^{1}\)1993; Zbl 0794.47034), (\({}^{2}\)2004; Zbl 1061.47001)], and “Nonlinear functional analysis. A first course” by S. Kesevan [(Texts and Readings in Mathematics 28, Hindustan Book Agency, New Delhi) (2004; Zbl 1054.46002)].

The book under review is a rather self-contained textbook, covering not only all classical topics of linear and nonlinear analysis, but focusing in particular on nonlinear problems arising in connection with ordinary and partial differential equations.

The first two chapters recall the topics of a typical course in linear algebra and (linear) functional analysis; most results are even proved. The authors discuss, in particular, complex and real Jordan normal forms, topological vector spaces, Sobolev spaces and the corresponding embedding theorems, the open mapping theorem, basic Hilbert space theory, duality theory, functional calculus, compact operators, and Riesz–Schauder theory. The Banach–Caccioppoli fixed point principle and its generalizations to nonexpansive maps are also treated in these chapters. The abstract results are illustrated by many standard examples involving integral operators or initial and boundary value problems.

Chapters 3 and 4 are concerned with integration and differentiation in infinite-dimensional spaces and their main applications. Both Riemann and Bochner integrals are treated, as well as Gâteaux and Fréchet derivatives. Even somewhat deeper topics like the symmetry of the second derivative without continuity hypothesis are treated. The implicit and inverse function theorems are discussed, including Hadamard’s global versions. Finally, numerous applications to (local) bifurcation theory are considered. The topics covered here include the Lyapunov–Schmidt reduction, the Morse normal form, and the Crandall–Rabinowitz bifurcation theorem.

The fifth chapter is devoted to topological and monotonicity methods. The chapter starts with Brouwer’s and Schauder’s fixed point theorems and their applications, but also fully develops the analytic approach for the topological Brouwer and Leray–Schauder degree in a very elegant way. Moreover, Krasnoselskij’s bifurcation theorem for odd multiplicities is treated. Finally, a certain version of Minty’s celebrated theorem on monotone operators (in a Hilbert space) is proved, and iteration methods in ordered Banach spaces are discussed.

Chapter 6 is about variational methods. After some introductory examples (containing also regularity results for global minima of typical integral functionals), the direct method of variation (i.e., concerning global minima) is discussed. The chapter proceeds with a section about Lagrange multipliers with applications to the Courant–Fisher minimax principle for eigenvalues in Hilbert spaces and to various boundary value problems. The chapter closes with the famous mountain pass lemma and saddle point theorems, explaining in detail the need and importance of the Palais–Smale condition.

The seventh and final chapter contains the most important applications, e.g., to the nonlinear partial differential equation \(\Delta u = g(x, u)\) with Dirichlet boundary values. The concept of weak solutions is carefully introduced, and the methods of the earlier chapters (Banach’s and Schauder’s fixed point theorems, as well as topological, monotonicity, and variational methods) are used to study them. This gives a fascinating insight into both the advantages and drawbacks of the various methods offered by nonlinear analysis, when one applies these methods to one and the same problem from different points of view.

The great merit of this book is that it leads the non-specialist into the very heart of nonlinear analysis, concentrating on what is actually needed, but without wasting too much time with technicalities. This is hence not an encyclopedic work, like some of the monographs mentioned above, but a genuine handbook “for the working analyst”. Almost every abstract result is motivated in advance and illustrated afterwards by some example or application and, vice versa, specific nonlinear problems are analyzed from the viewpoint of finding the “appropriate” method to solve them. As a result, this book can be recommended to any student of mathematics or to any scientist or engineer interested in nonlinear problems, having some basic knowledge in linear algebra and calculus. It is to be hoped that the book will find the large readership it deserves.

In the past five years, however, an increasing interest in nonlinear problems led to the publication of other monographs, devoted in part to special theoretical aspects, in part to more application-oriented topics in the field. Without aiming for a complete coverage, of course, we mention here “Elements of nonlinear analysis” by M. Chipot [(Birkhäuser, Basel) (2000; Zbl 0964.35002)], “Geometric nonlinear functional analysis” by Y. Benyamini and J. Lindenstrauss [(AMS, Providence/RI) (2000; Zbl 0946.46002)], “Nonlinear functional analysis” by W. Takahashi [(Yokohama Publ.) (2000; Zbl 0997.47002)], “Spectral theory and nonlinear functional analysis” by J. López–Gómez [(Chapman & Hall, Boca Raton/FL) (2001; Zbl 0978.47048)], “An introduction to nonlinear analysis” by Z. Denkowski, S. Migórski and N. S. Papageorgiou [(Kluwer, Dordrecht) (2003; Zbl 1040.46001)], “Nichtlineare Funktionalanalysis. Eine Einführung” by M. Růžička [(Springer, Berlin) (2004; Zbl 1043.46002)], “A topological introduction to nonlinear analysis” by R. F. Brown [(Birkhäuser, Boston/MA) (\({}^{1}\)1993; Zbl 0794.47034), (\({}^{2}\)2004; Zbl 1061.47001)], and “Nonlinear functional analysis. A first course” by S. Kesevan [(Texts and Readings in Mathematics 28, Hindustan Book Agency, New Delhi) (2004; Zbl 1054.46002)].

The book under review is a rather self-contained textbook, covering not only all classical topics of linear and nonlinear analysis, but focusing in particular on nonlinear problems arising in connection with ordinary and partial differential equations.

The first two chapters recall the topics of a typical course in linear algebra and (linear) functional analysis; most results are even proved. The authors discuss, in particular, complex and real Jordan normal forms, topological vector spaces, Sobolev spaces and the corresponding embedding theorems, the open mapping theorem, basic Hilbert space theory, duality theory, functional calculus, compact operators, and Riesz–Schauder theory. The Banach–Caccioppoli fixed point principle and its generalizations to nonexpansive maps are also treated in these chapters. The abstract results are illustrated by many standard examples involving integral operators or initial and boundary value problems.

Chapters 3 and 4 are concerned with integration and differentiation in infinite-dimensional spaces and their main applications. Both Riemann and Bochner integrals are treated, as well as Gâteaux and Fréchet derivatives. Even somewhat deeper topics like the symmetry of the second derivative without continuity hypothesis are treated. The implicit and inverse function theorems are discussed, including Hadamard’s global versions. Finally, numerous applications to (local) bifurcation theory are considered. The topics covered here include the Lyapunov–Schmidt reduction, the Morse normal form, and the Crandall–Rabinowitz bifurcation theorem.

The fifth chapter is devoted to topological and monotonicity methods. The chapter starts with Brouwer’s and Schauder’s fixed point theorems and their applications, but also fully develops the analytic approach for the topological Brouwer and Leray–Schauder degree in a very elegant way. Moreover, Krasnoselskij’s bifurcation theorem for odd multiplicities is treated. Finally, a certain version of Minty’s celebrated theorem on monotone operators (in a Hilbert space) is proved, and iteration methods in ordered Banach spaces are discussed.

Chapter 6 is about variational methods. After some introductory examples (containing also regularity results for global minima of typical integral functionals), the direct method of variation (i.e., concerning global minima) is discussed. The chapter proceeds with a section about Lagrange multipliers with applications to the Courant–Fisher minimax principle for eigenvalues in Hilbert spaces and to various boundary value problems. The chapter closes with the famous mountain pass lemma and saddle point theorems, explaining in detail the need and importance of the Palais–Smale condition.

The seventh and final chapter contains the most important applications, e.g., to the nonlinear partial differential equation \(\Delta u = g(x, u)\) with Dirichlet boundary values. The concept of weak solutions is carefully introduced, and the methods of the earlier chapters (Banach’s and Schauder’s fixed point theorems, as well as topological, monotonicity, and variational methods) are used to study them. This gives a fascinating insight into both the advantages and drawbacks of the various methods offered by nonlinear analysis, when one applies these methods to one and the same problem from different points of view.

The great merit of this book is that it leads the non-specialist into the very heart of nonlinear analysis, concentrating on what is actually needed, but without wasting too much time with technicalities. This is hence not an encyclopedic work, like some of the monographs mentioned above, but a genuine handbook “for the working analyst”. Almost every abstract result is motivated in advance and illustrated afterwards by some example or application and, vice versa, specific nonlinear problems are analyzed from the viewpoint of finding the “appropriate” method to solve them. As a result, this book can be recommended to any student of mathematics or to any scientist or engineer interested in nonlinear problems, having some basic knowledge in linear algebra and calculus. It is to be hoped that the book will find the large readership it deserves.

Reviewer: Jürgen Appell (Würzburg)

##### MSC:

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

47Hxx | Nonlinear operators and their properties |

37Nxx | Applications of dynamical systems |

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