×

Nonlinear semigroups. Translated from the Japanese by Choong Yun Cho. (English) Zbl 0766.47039

Translations of Mathematical Monographs. 109. Providence, RI: American Mathematical Society (AMS). viii, 230 p. (1992).
This book treats the basic theory of semigroups of nonlinear contractions in a Banach space and some applications to the Cauchy problem for nonlinear evolution equations.
Chapter I is a good and clear summary of the main results on Functional Analysis which are used in subsequent chapters; the results are given without proofs but with appropriate general references.
Chapter II, which begins with a detailed study of the duality mapping in uniformly convex spaces, is dedicated to the study of the main properties of dissipative, \(m\)-dissipative and maximal dissipative operators. It is established the relation between maximal dissipative operators and \(m\)- dissipative operators in Hilbert spaces and with the help of an example the author shows that it is not true in general Banach spaces.
In Chapter III the author considers the relation between semigroups and dissipative operators when a semigroup of contractions is given in a Banach space \(X\). Some connections between the infinitesimal generator, weak infinitesimal generator and \((g)\)-operator of the semigroup are established in strictly convex and uniformly convex spaces. In what follows, he studies the problem about the existence of the infinitesimal generator, specially in the case of Hilbert spaces.
In Chapter IV the author deals with the problem about generation by dissipative operators of semigroups of contractions and its relation with the infinitesimal generator. Elementary applications to some Cauchy problems are also given. Next, it is proved a characterization of the infinitesimal generator of a semigroup of contractions when the space \(X\) and its conjugate are uniformly convex.
Chapter V deals with the Cauchy problem for nonlinear evolution equations, showing conditions for the existence and uniqueness of (integral) solutions. Also, the convergence of the difference approximations for the Cauchy problem are discussed.
In Chapter VI the author studies the theory of convergence and approximation of semigroups and the problem whether \(A+B\) generates a semigroup or not when \(A\) is a dissipative operator generating a semigroup and \(B\) is a continuous operator. Some applications of these results to evolution equations are also presented.
The last chapter is dedicated to the application of some previous results to the study of Cauchy problems for quasilinear partial differential equations of first order. Moreover, the book contains an appendix about a minimax Theorem and a postcript covering the description of the principal references for each chapter and related items.
It must be remarked that throughout the material the author shows some similarities and differences of the nonlinear theory with respect to the linear one. The emphasis is laid on the theoretical part and only about twenty examples are exhibited to illustrate the theoretical concept and results. Theorems are proved with care and a reader who has knowledge in Functional Analysis at the level of a college junior may understand the content of the book. In short, the book is a good source of information for all those interested in the theory of nonlinear semigroups.

MSC:

47H20 Semigroups of nonlinear operators
47H06 Nonlinear accretive operators, dissipative operators, etc.
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47N20 Applications of operator theory to differential and integral equations
47F05 General theory of partial differential operators
58D07 Groups and semigroups of nonlinear operators
PDFBibTeX XMLCite