Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

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

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

BIOGRAPHIES:

Behzad Djafari Rouhani received his PhD degree from Yale University in 1981, under the direction of the late Professor Shizuo Kakutani. He is currently a Professor of Mathematics at the University of Texas at El Paso, USA.

Hadi Khatibzadeh received his PhD degree form Tarbiat Modares University in 2007, under the direction of the first author. He is currently an Associate Professor of Mathematics at University of Zanjan, Iran.

They both work in the field of Nonlinear Analysis and its Applications, and they each have over 50 refereed publications.

Narcisa Apreutesei