Project Materials

Monotone Operators and Applications

TABLE OF CONTENTS

Preliminaries 7
1.1 Geometry of Banach Spaces . . . . . . . . . . . . . . . . . . . 7
1.1.1 Uniformly Convex Spaces . . . . . . . . . . . . . . . . 7
1.1.2 Strictly Convex Spaces . . . . . . . . . . . . . . . . . . 9
1.1.3 Duality Mappings. . . . . . . . . . . . . . . . . . . . 10
1.1.4 Duality maps of Lp Spaces (p > 1) . . . . . . . . . . . 13
1.2 Convex Functions and Sub-differentials . . . . . . . . . . . . . 15
1.2.1 Basic notions of Convex Analysis . . . . . . . . . . . . 15
1.2.2 Sub-differential of a Convex function . . . . . . . . . . 19
1.2.3 Jordan Von Neumann Theorem for the Existence of Saddle point . . . 20
2 Monotone operators. Maximal monotone operators. 23
2.1 Maximal monotone operators . . . . . . . . . . . . . . . . . . 23
2.1.1 Definitions, Examples and properties of Monotone Operators . . 23
2.1.2 Rockafellar’s Characterization of Maximal Monotone Operators . . . 27
2.1.3 Topological Conditions for Maximal Monotone Operators . . . 35
2.2 The sum of two maximal monotone operators . . . . . . . . . 37
2.2.1 Resolvent and Yosida Approximations of Maximal Monotone Operators . 37
2.2.2 Basic Properties of Yosida Approximations . . . . . . 38
3 On the Characterization of Maximal Monotone Operators 46
3.1 Rockafellar’s characterization of maximal monotone operators. 46
4 Applications 51
4.1 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Uniformly Monotone Operators . . . . . . . . . . . . . . . . . 52

CHAPTER ONE

Preliminaries

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; xi the value of the function x 2 X at x 2 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 kk 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 Xw

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 ” 2 (0; 2] there exist a = (“) > 0 such that for each x; y 2 X with kxk 1, kyk 1, and kx 􀀀 yk “, we have k1