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Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

Authors
Publisher Apple Academic Press Inc.
Year 2019
Pages 240
Version hardback
Readership level General/trade
Language English
ISBN 9781482228182
Categories Functional analysis & transforms
$157.07 (with VAT)
698.25 PLN / €149.70 / £129.96
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Book description

This book is devoted to the study of nonlinear evolution and difference equations of first and second order governed by a maximal monotone operator. This class of abstract evolution equations contains not only a class of ordinary differential equations, but also unify some important partial differential equations, such as the heat equation, wave equation, Schrodinger equation, etc.





In addition to their applications in ordinary and partial differential equations, this class of evolution equations and their discrete version of difference equations have found many applications in optimization.





In recent years, extensive studies have been conducted in the existence and asymptotic behaviour of solutions to this class of evolution and difference equations, including some of the authors works. This book contains a collection of such works, and its applications.





Key selling features:




Discusses in detail the study of non-linear evolution and difference equations governed by maximal monotone operator
Information is provided in a clear and simple manner, making it accessible to graduate students and scientists with little or no background in the subject material
Includes a vast collection of the authors' own work in the field and their applications, as well as research from other experts in this area of study

Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

Table of contents

Table of Contents:











PART I. PRELIMINARIES











Preliminaries of Functional Analysis





Introduction to Hilbert Spaces





Weak Topology and Weak Convergence





Reexive Banach Spaces





Distributions and Sobolev Spaces











Convex Analysis and Subdifferential Operators





Introduction





Convex Sets and Convex Functions





Continuity of Convex Functions





Minimization Properties





Fenchel Subdifferential





The Fenchel Conjugate











Maximal Monotone Operators





Introduction





Monotone Operators





Maximal Monotonicity





Resolvent and Yosida Approximation





Canonical Extension











PART II - EVOLUTION EQUATIONS OF MONOTONE TYPE











First Order Evolution Equations





Introduction





Existence and Uniqueness of Solutions





Periodic Forcing





Nonexpansive Semigroup Generated by a Maximal Monotone Operator





Ergodic Theorems for Nonexpansive Sequences and Curves





Weak Convergence of Solutions and Means





Almost Orbits





Sub-differential and Non-expansive Cases





Strong Ergodic Convergence





Strong Convergence of Solutions





Quasi-convex Case











Second Order Evolution Equations





Introduction





Existence and Uniqueness of Solutions





Two Point Boundary Value Problems





Existence of Solutions for the Nonhomogeneous Case





Periodic Forcing





Square Root of a Maximal Monotone Operator





Asymptotic Behavior





Asymptotic Behavior for some Special Nonhomogeneous Cases























Heavy Ball with Friction Dynamical System





Introduction





Minimization Properties











PART III. DIFFERENCE EQUATIONS OF MONOTONE TYPE











First Order Difference Equations and Proximal Point Algorithm





Introduction





Boundedness of Solutions





Periodic Forcing





Convergence of the Proximal Point Algorithm





Convergence with Non-summable Errors





Rate of Convergence











Second Order Difference Equations





Introduction





Existence and Uniqueness





Periodic Forcing





Continuous Dependence on Initial Conditions





Asymptotic Behavior for the Homogeneous Case





Subdifferential Case





Asymptotic Behavior for the Non-Homogeneous Case





Applications to Optimization











Discrete Nonlinear Oscillator Dynamical System and the Inertial Proximal Algorithm





Introduction





Boundedness of the Sequence and an Ergodic Theorem





Weak Convergence of the Algorithm with Errors





Subdifferential Case





Strong Convergence











PART IV. APPLICATIONS





Some Applications to Nonlinear Partial Differential Equations and Optimization





Introduction





Applications to Convex Minimization and Monotone Operators





Application to Variational Problems





Some Applications to Partial Differential Equations











Complete Bibliography

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