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Viser: Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces
Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces Vital Source e-bog
Behzad Djafari Rouhani
(2019)
Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces
Behzad Djafari Rouhani, Narcisa Apreutesei og Hadi Khatibzadeh
(2019)
Sprog: Engelsk
om ca. 10 hverdage
Detaljer om varen
- 1. Udgave
- Vital Source searchable e-book (Reflowable pages)
- Udgiver: CRC Press (Maj 2019)
- ISBN: 9780429528880
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Detaljer om varen
- Hardback: 240 sider
- Udgiver: CRC Press LLC (Marts 2019)
- Forfattere: Behzad Djafari Rouhani, Narcisa Apreutesei og Hadi Khatibzadeh
- ISBN: 9781482228182
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
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 >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 xistence 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 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 t;/P>
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