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
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Für höhere Schule und Studium
Für Beruf und Forschung
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Höhe: 240 mm
Breite: 161 mm
Dicke: 18 mm
Gewicht
ISBN-13
978-1-4822-2818-2 (9781482228182)
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Schweitzer Klassifikation
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
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
