Nonlinear Semigroups and Differential Equations in Banach Spaces

ISBN-13: 9789028602052 / Angielski / Miękka / 1976 / 352 str.

Viorel Barbu

Opis

This book is concerned with nonlinear semigroups of contractions in Banach spaces and their application to the existence theory for differential equa- tions associated with nonlinear dissipative operators. The study of nonlinear semi groups resulted from the examination of nonlinear parabolic equations and from various nonlinear boundary value problems. The first work done by Y. Komura stimulated much further work and interest in this subject. Thus a series of studies was begun and then continued by T. Kato, M. G. Crandall, A. Pazy, H. Brezis and others, who made important con- tributions to the development of the theory. The theory as developed below is a generalisation of the Hille-Yosida theory for one-parameter semigroups of linear operators and is a collection of diversified results unified more or less loosely by their methods of approach. This theory is also closely related to the theory of nonlinear monotone operators. Of course not all aspects of this theory could be covered in our expo- sition, and many important contributions to the subject have been excluded for the sake of brevity. We have attempted to present the basic results to the reader and to orient him toward some of the applications. This book is intended to be self-contained. The reader is assumed to have only a basic knowledge of functional analysis, function theory and partial differential equations. Some of the necessary prerequisites for the reading of this 'book are summarized, with or without proof, in Chapter I.

Spis treści

I Preliminaries.- § 1. Metric properties of normed spaces.- 1.1 Duality mappings.- 1.2 Strictly convex normed spaces.- 1.3 Uniformly convex Banach spaces.- § 2. Vectorial functions defined on real intervals.- 2.1 Absolutely continuous vectorial functions.- 2.2 Vectorial distributions and Wk,p spaces.- 2.3 Sobolev spaces.- § 3. Semigroups of continuous linear operators.- 3.1 Semigroups of class (C0). Hille-Yosida theorem.- 3.2 Analytic semigroups.- 3.3 Nonhomogeneous linear differential equations.- II Nonlinear Operators in Banach Spaces.- § 1. Maximal monotone operators.- 1.1 Definitions and fundamental concepts.- 1.2 A general perturbation theorem.- 1.3 A nonlinear elliptic boundary problem.- § 2. Subdifferential mappings.- 2.1 Lower semicontinuous convex functions.- 2.2 Subdifferentials of convex functions.- 2.3 Some examples of cyclically monotone operators.- § 3. Dissipative sets in Banach spaces.- 3.1 Basic properties of dissipative sets.- 3.2 Perturbations of dissipative sets.- 3.3 Riccati equations in Hilbert spaces.- Bibliographical notes.- III Differential Equations in Banach Spaces.- § 1. Semigroups of nonlinear contractions in Banach spaces.- 1.1 General properties of nonlinear semigroups.- 1.2 The exponential formula.- 1.3 Convergence theorems.- 1.4 Generation of nonlinear semigroups.- § 2. Quasi-autonomous differential equations.- 2.1 Existence theorems.- 2.2 Periodic solutions.- 2.3 Examples.- § 3. Differential equations associated with continuous dissipative operators.- 3.1 A general existence result.- 3.2 Continuous perturbations of m-dissipative operators.- 3.3 Semi-linear second-order elliptic equations in L1.- § 4. Time-dependent nonlinear differential equations.- 4.1 Evolution equations associated with dissipative sets.- 4.2 Evolution equations associated with nonlinear monotone hemicon-tinuous operators.- Bibliographical notes.- IV Nonlinear Differential Equations in Hilbert Spaces.- § 1. Nonlinear semigroups in Hilbert spaces.- 1.1 Nonlinear version of the Hille-Yosida theorem.- 1.2 Exponential formulae.- 1.3 Invariant sets with respect to nonlinear semigroups.- § 2. Smoothing effect on initial data.- 2.1 The case in which A = ? ?.- 2.2 The case in which int D(A) ? ?.- 2.3 Applications.- § 3. Variational evolution inequations.- 3.1 Unilateral conditions on u(t).- 3.2 Unilateral conditions on $$ \frac{}{}(t) $$.- 3.3 A class of nonlinear variational inequations.- 3.4 Applications.- § 4. Nonlinear Volterra equations with positive kernels in Hilbert spaces.- 4.1 Positive kernels.- 4.2 Equation (4.1) with A = ? ?.- 4.3 Equation (4.1) with A demicontinuous.- 4.4 A class of integro-differential equations.- 4.5 Further investigation of the preceding case.- Bibliographical notes.- V Second Order Nonlinear Differential Equations.- § 1. Nonlinear differential equations of hyperbolic type.- 1.1 The equation $$ \frac{{{d^2}u}}{{d{t^2}}} + Au + M\left( {\frac{}{}} \right) \mathrel\backepsilon f $$.- 1.2 Further investigation of the preceding case.- 1.3 Examples.- 1.4 Singular perturbations and hyperbolic variational inequations.- 1.5 Nonlinear wave equation.- § 2. Boundary value problems for second order nonlinear differential equations.- 2.1 A class of two-point boundary value problems.- 2.2 Examples.- 2.3 A boundary value problem on half-axis.- 2.4 The square root of a nonlinear maximal monotone operator.- Bibliographical notes.