Nonlinear Evolution and Difference Equations of Monotone Type in Hilbert Spaces

PRELIMINARIES. Preliminaries of Functional Analysis. Introduction to Hilbert Spaces. Weak Topology and Weak Convergence. Sobolev Spaces. Convex Functions And Subdifferentials. Continuity Properties of Convex Functions. Minimization of Convex Functionals. Subdifferential Mappings. Maximal Monotone Operators. Monotone Operators and Their Properties. Maximal Monotonicity. Resolvent and Yosida Approximation. EVOLUTION EQUATIONS OF MONOTONE TYPE. First Order Evolution Equations. Existence and Uniqueness of Solutions. Periodic Forcing. Nonexpansive Semigroup Generated by a Maximal Monotone Operator. Nonexpansive and Almost Nonexpansive Sequences and Curves. Ergodic Convergence of Solutions. Weak and Strong Convergence. Subdifferential Case. Second Order Evolution Equations. Existence and Uniqueness of Solutions. Periodic Forcing. Square Root of a Maximal Monotone Operator. Continuous Dependence on Data. Ergodic Convergence of Solutions. Weak and Strong Convergence. Sub differential Case. Nonlinear Oscillator Dynamical System. Modeling the Oscillation of a Heavy Ball with Friction. Minimization Properties. Quasi-Convex and Non-Convex Cases. DIFFERENCE EQUATIONS OF MONOTONE TYPE (DISCRETE VERSIONS). First Order Difference Equations and Proximal point Algorithm. Introduction. Boundedness of Solutions. Periodic Forcing. Convergence of the Proximal Point Algorithm. More General Errors. Rate of Convergence. Second Order Difference Equations. Introduction. Existence and Uniqueness. Periodic Forcing. Continuous Dependence on Data. Ergodic Convergence. Weak and Strong Convergence. Applications to Optimization. Discrete Nonlinear Oscillator Dynamical System And Inertial Proximal Algorithm. Introduction. Weak and Strong Convergence. Subdifferential Case. Quasi-Convex and Non-Convex Cases. APPLICATIONS. Some Applications to Nonlinear Partial Differential equations and Optimization.