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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

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