Leyden, Noordhoff International Publishing, 1976, 352p, ISBN 90-286-0205-4.The book is concerned with nonlinear semigroups of contractions in Banach spaces and their application to the existence theory for differential equations associated with nonlinear dissipative operators.Table of content:
Preface to the first edition
Preface to the English edition
Chapter I. Preliminaries
Metric properties of normed spaces
Duality mappings
Strictly convex normed spaces
Uniformly convex Banach spaces
Vectorial functions defined on real intervals
Absolutely continuous vectorial functions
Vectorial distributions and W^(k,p) spaces
Sobofev spaces
Semigroups of continuous linear operators
Semigroups of class (C0). Hille-Vosida theorem
Analytic semigroups
Nonhomogeneous linear differential equations
Chapter II. Nonlinear operators in banach spaces
Maximal monotone operators
Definitions and fundamental concepts
A general perturbation theorem
A nonlinear elliptic boundary problem
Subdifferential mappings
Lower semicontinuous convex functions
Subdifferentials of convex functions
Some examples of cyclically monotone operators
Dissipative sets in Banach spaces
Basic properties of dissipative sets
Perturbations of dissipative sets
Riccati equations in Hilbert spaces 89
Bibliographical notes
Chapter III. Differential equations in banach spaces
Semigroups of nonlinear contractions in Banach spaces
General properties of nonlinear semigroups
The exponential formula
Convergence theorems
Generation of nonlinear semigroups
Quasi-autonomous differential equations
Existence theorems
Periodic solutions
Examples
Differential equations associated with continuous dissipative operators
A general existence result
Continuous perturbations of m-dissipative operators
Semi-linear second order elliptic equations in L1
Time-dependent nonlinear differential equations
Evolution equations associated with dissipative sets
Evolution equations associated with nonlinear monotone hemicontinuous operators
Bibliographical notes
Chapter IV. Nonlinear differential equations in hilbert spaces
Nonlinear semigroups in Hilheri spaces
Nonlinear version of the Hille-Yosida theorem
Exponential formulae
Invariant sets with respect to nonlinear semigroups
Smoothing effect on initial data
The case in which D(A)= pd phi
The case in which int D(A) is not empty
Applications
Variational evolution inequations
Unilateral conditions on u(t)
Unilateral conditions on d u(t)/dt
A class of nonlinear variational inequations
Applications
Nonlinear Volterra equations with positive kernels in Hilbert spaces
Positive kernels
Equation (4.1) with A = pd phi
Equation (4.1) with A demicontinuous
A class of integro-differential equations
Further investigation of the preceding case 261
Bibliographical notes
Chapter V. Second order nonlinear differential equations
Nonlinear differential equations of hyperbolic type
The equation d^2 u/dt^2 + Au + M(du/dt) contains f
Further investigation of the preceding case
Examples
Singular perturbations and hyperbolic variational inequations
Nonlinear wave equation
Boundary value problems for second order nonlinear differential equations
A class of two-point boundary value problems
Examples
A boundary value problem on half-axis
The square root of a nonlinear maximal monotone operator
Bibliographical notes
Bibliography
Subject index
