# Project Materials

## Monotone Operators and Applications

TABLE OF S

Preliminaries 7
1.1 Geometry of Banach s . . . . . . . . . . . . . . . . . . . 7
1.1.1 Uniformly Convex s . . . . . . . . . . . . . . . . 7
1.1.2 Strictly Convex s . . . . . . . . . . . . . . . . . . 9
1.1.3 Duality Mappings. . . . . . . . . . . . . . . . . . . . 10
1.1.4 Duality maps of Lp s (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 for the 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 s . . 23
2.1.2 Rockafellar’s Characterization of Maximal Monotone s . . . 27
2.1.3 ological Conditions for Maximal Monotone s . . . 35
2.2 The sum of two maximal monotone operators . . . . . . . . . 37
2.2.1 Resolvent and Yosida s of Maximal Monotone s . 37
2.2.2 Basic Properties of Yosida s . . . . . . 38
3 On the Characterization of Maximal Monotone s 46
3.1 Rockafellar’s characterization of maximal monotone operators. 46
4 Applications 51
4.1 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 Uniformly Monotone s . . . . . . . . . . . . . . . . . 52

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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 s

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 s

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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