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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

Autorzy
Wydawnictwo Apple
Data wydania 01/03/2019
Wydanie Pierwsze
Liczba stron 240
Forma publikacji książka w twardej oprawie
Poziom zaawansowania Literatura popularna
Język angielski
ISBN 9781482228182
Kategorie Analizy funkcjonalne i przekształcenia, Rachunek różniczkowy i formuły
549.05 PLN (z VAT)
$141.18 / €126.32 / £117.00 /
Produkt na zamówienie
Przesyłka w 3-4 tygodnie
Ilość
Do schowka

Opis książki

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

Spis treści

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

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