Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

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