Monotone Operators and Applications

Description

– Monotone Operators and Applications –

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Abstract

This project is mainly focused on the theory of Monotone (increasing) Operators and its applications. Monotone operators play an important role in many branches of Mathematics such as Convex Analysis, Optimization Theory, Evolution Equations Theory, Variational Methods and Variational Inequalities.

Basic examples of monotone operators are positive semi-definite matrices A of order n ∈ N (since they define linear operators on R n and satisfy hAx, xi ≥ 0 for all x ∈ R n ), projection operators pC onto closed convex nonempty subsets C of a Hilbert space (since hx − y, pC (x) − pC (y)i ≥ 0 for all x, y ∈ H),

the derivative Df of a differentiable convex function f defined in a Banach space (since hx − y, Df(x) − Df(y)i ≥ 0 for all x, y ∈ dom(f)), and the elliptic differential operator −∆ on H2 (R n ).

Monotone operators which have no proper monotone extension are called maximal monotone operators and are of particular interest because they are crucial in the solvability of evolution equations in Hilbert spaces as they generate semigroup of bounded linear operators. 

Introduction

1.1 Background of the Study

The aim of this chapter is to provide some basic results pertaining to geometric properties of normed linear spaces and convex functions. Some of these results, which can be easily found in textbooks are given without proofs or with a sketch of proof only.  

1.1 Geometry of Banach Spaces  

Throughout this chapter X denotes a real norm space and X∗ denotes its corresponding dual. We shall denote by the pairing hx, x∗ i the value of the function x ∗ ∈ X∗ at x ∈ X.

The norm in X is denoted by k · k, while the norm in X∗ is denoted by k · k∗. If there is no danger of confusion, we omit the asterisk from the notation k· k∗ and denote both norm in X and X∗ by the symbol k · k.

As usual We shall use the symbol → and * to indicate strong and weak convergence in X and X∗ respectively. We shall also use w ∗ –lim to indicate the weak-star convergence in X∗. The space X∗ endowed with the weak-star topology is denoted by X∗ w 

1.1.1 Uniformly Convex Spaces  

Definition 1.1. Let X be a normed linear space. Then X is said to be uniformly convex if for any ε ∈ (0, 2] there exist a δ = δ(ε) > 0 such that for each x, y ∈ X with kxk ≤ 1, kyk ≤ 1, and kx − yk ≥ ε, we have k 1 2 (x + y)k ≤ 1 − δ.  

Theorem 1.2. Let X be a uniformly convex space. Then for any d > 0, ε > 0 and x, y ∈ X with kxk ≤ d, kyk ≤ d, and kx − yk ≥ ε, there exist a δ = δ( ε d) > 0 such that k 1 2 (x + y)k ≤ (1 − δ)d.

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