1st Edition

Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

    248 Pages
    by CRC Press

    248 Pages
    by CRC Press

    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



    Biography

    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