# Monotone Operators and Applications

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**Monotone Operators and Applications.**

### PREFACE

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.

### TABLE OF CONTENTS

1 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 L

p Spaces (p > 1) . . . . . . . . . . . 13

1.2 Convex Functions and Subdifferentials . . . . . . . . . . . . . 15

1.2.1 Basic notions of Convex Analysis . . . . . . . . . . . . 15

1.2.2 Subdifferential 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

### 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, 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.

### BIBLIOGRAPHY

V. Barbu, Analysis and control of Nonlinear Infinite Dimensionl

Systems, School of Mathematics, University of Iasi, Iasi Romania,

November 1991.

V. Barbu, Nonlinear Differential Equations of Monotone types in

Banach Spaces, School of Mathematics, University of Iasi, Iasi Romania, September 2009.

C.E Chidume, Geometric Properties of Banach Spaces and Nonlinear Iterations, International Center for Theoretical Physics, Triesty,

Italy, June 2008.

C.E Chidume, Applicable Functional Analysis, International Center

for Theoretical Physics, Triesty, Italy, July 2006.

E. Ziedler, Nonlinear functional analysis and its applications, Part

II: Monotone operators (springer-Verlag, Berlin/New York, 1985).

K. Ito and F. Kappel, Evolution Equations and Approximations,

North California University, USA, December 2001.

S. Simons and C. Zalinescu, A New Proof for Rockerfellar’s Characterizations of Maximal Monotone Operators, Proc. Amer. math. Soc. 131(2004), pp. 2969-2972.

H. Brezis, Monotone Operators, Nonlinear Semigroups and applications, Proc.. of the Int. Cong. of Math. Vancouver, (1974), 249-255.

A. Besenyei, On Uniformly Monotone Operators arising in Nonlinear Elliptic and Parabolic Problems, Annales Univ. Sci. Budapest.Etovos Sect. Math. (2011).

J. Leray and Jacques-L. Lions, Quelques results de Visik sur les

problemes elliptiques non lineaires par les methodes de Minty Browder Bulletin de la S.M.F, tome 93 (1965), p.97-107.

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