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GENERAL

Monotone Operators and Applications – Premium Researchers

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and

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 operators. Maximal monotone operators. 23
2.1 Maximal monotone operators . . . . . . . . . . . . . . . . . . 23
2.1.1 Definitions, Examples and properties of . . 23
2.1.2 . . . 27
2.1.3 Topological Conditions for . . . 35
2.2 The sum of two maximal monotone operators . . . . . . . . . 37
2.2.1 Resolvent and of . 37
2.2.2 Basic Properties of . . . . . . 38
3 On the Characterization of 46
3.1 Rockafellar’s characterization of maximal monotone operators. 46
4 51
4.1 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Uniformly . . . . . . . . . . . . . . . . . 52

CHAPTER ONE

Preliminaries

The aim of this chapter is to provide some basic results pertaining to of normed and .

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

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