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Maximal Monotone Operators in Banach Spaces

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Maximal Monotone Operators in Banach Spaces Maximal Monotone Operators in Banach Spaces B. A. C. S. Balasuriya This thesis is presented for the degree of Master of Science of The University of Western Australia The School of Mathematics and Statistics 2004 This thesis is dedicated to My Family and to the memory of Simon Abstract Our aim in this research was to study monotone operators in Banach spaces. In particular, the most important concept in this theory, the maximal mono- tone operators. Here we make an attempt to describe most of the important results and concepts on maximal monotone operators and how they all tie together. We will take a brief look at subdifferentials, which generalize the notion of a derivative. The subdifferential is a maximal monotone operator and it has proved to be of fundamental importance for the study of maximal monotone operators. The theory of maximal monotone operators is some- what complete in reflexive Banach spaces. However, in nonreflexive Banach spaces it is still to be developed fully. As such, here we will describe most of the important results about maximal monotone operators in Banach spaces and we will distinguish between the reflexive Banach spaces and nonreflexive Banach spaces when a property is known to hold only in reflexive Banach spaces. In the latter case, we will state what the corresponding situation is in nonreflexive Banach spaces and we will give counter examples whenever such a result is known to fail in nonreflexive Banach spaces. The representa- tions of monotone operators by convex functions have found to be extremely useful for the study of maximal monotone operators and it has generated a lot of interest of late. We will discuss some of those key representations and their properties. We will also demonstrate how these representations could be utilized to obtain results about maximal monotone operators. We have included a discussion about the very important Rockafellar sum theorem and some its generalizations. This key result and its generalizations have only been proved in reflexive Banach spaces. We will also discuss several special cases where the Rockafellar sum theorem is known to be true in nonreflex- ive Banach spaces. The subclasses which provide a basis for the study of monotone operators in nonreflexive Banach spaces are also discussed here. Contents Acknowledgement . iii Introduction . v 0 Tools 1 0.1 The Hahn-Banach Theorem . 2 0.2 Convex Functions and Semicontinuity . 4 0.3 The Minimax Theorem . 6 0.4 Baire Category Theorem . 9 1 Monotone Operators 13 1.1 Monotonicity . 13 1.2 Local Boundedness of Monotone Operators . 15 1.3 Maximal Monotone Operators . 18 2 Subdifferentials 23 2.1 Subdifferential of a Sum . 24 2.2 Rockafellar’s Maximal Monotonicity Theorem . 28 3 More on Maximal Monotone Operators 37 3.1 Maximal Monotonicity in Reflexive Banach Spaces . 37 3.2 Surjectivity of Maximal Monotone Operators . 39 3.3 Convexity of Domains and Ranges . 43 3.4 Yosida Approximations . 49 4 Representations of a Monotone Operator 51 4.1 A Monotone Operator Defined by a Convex Function . 52 i 4.2 The Fitzpatrick Function . 55 4.3 The χ and ψ˜ Functions of Coodey-Simons . 60 5 Sum Theorem in Reflexive Banach Spaces 63 5.1 Constraint Qualifications . 63 5.2 Rockafellar’s Sum Theorem . 65 5.3 Generalizations of the Sum Theorem . 71 6 Subclasses of Maximal Monotone Operators 83 6.1 Maximal Monotone Operators of Type D ............ 84 6.2 Monotone Operators of Type FP ................ 88 6.3 Monotone Operators of Type FPV . 100 6.4 Regular Maximal Monotone Operators . 103 6.5 More Subclasses . 109 6.6 Relationships Among the Subclasses . 111 7 Sum Theorem in Nonreflexive Banach Spaces 113 7.1 Rockafellar’s Sum Theorem . 113 7.2 The Verona-Verona Result . 127 Some Concluding Remarks 131 Bibliography 133 iii Acknowledgement I would like to express my gratitude to my supervisor, Dr Simon Fitzpatrick for his invaluable advice and guidance throughout this research. I am ex- tremely grateful for his help and support without which this thesis would not have been possible. I would like to thank Dr Jonathan Borwein, Dr Jean-Paul Penot, Dr Stephen Simons and Dr Constantin Zalinescu for allowing me to use their unpublished material in my thesis. I am also thankful to Dr Grant Keady for reading a draft of this thesis and making several helpful comments. I am also grateful to all those who helped me through various means during my stay in Perth. v Introduction The monotone operators have found to be quite useful in several important areas of Mathematics such as Partial Differential Equations, Operator The- ory, and Numerical Analysis. This is because the concept of a monotone operator is general enough to cover subdifferentials and continuous positive linear operators, two objects which are so often found in the above areas. It is debatable as to who introduced the concept of monotone operators. However, the popular view is that, M. Golomb was the one who first intro- duced the monotone operators in a paper titled Zur Theorie der nichtlinearen Integralgleichungen, Integraleichungssysteme und allgemeinen Funktionalgle- ichungen in 1935. One of the earliest papers to consider the applications of monotone operators was Solving functional equations by contractive averag- ing by E. H. Zarantonello. Detailed discussions on the history of monotone operators can be found in J. Minty [21], E. Zeidler [48]. The modern theory of monotone operators was initiated independently by a series of papers by Minty and F. E. Browder in 1962. Later on, the ideas in these papers were extended considerably by R. T. Rockafellar to develop the theory as we know it now. Naturally, the progression of the theory of monotone operators was from the Hilbert spaces to reflexive Banach spaces and finally, to nonreflexive Banach spaces. Whilst the theory of maximal monotone operators in reflexive Banach spaces is somewhat complete, most of the key results remain unproved in the nonreflexive case. Our goal in this research was to study monotone operators in Banach spaces. In particular, the most important concept in the theory of monotone opera- tors, the maximal monotone operators. A brief outline of the material we present in thesis can be described in the following way. In Chapter 0 we will discuss the fundamental results from Functional Anal- ysis and Convex Analysis that we will need in the chapters to follow. We will introduce the monotone operators and the concept of maximal mono- tonicity in Chapter 1. Here we will also discuss the local boundedness of monotone operators, a quite useful notion in this theory. vi Chapter 2 is about the subdifferentials, an important class of monotone op- erators. We will discuss the main properties of subdifferentials and we will show that the subdifferential is a maximal monotone operator. The other key result of this Chapter is that under suitable conditions the subdifferen- tial of a sum is the sum of the subdifferentials. Subdifferentials play a central part in the monotone operator theory as they act as prototypes when general questions about maximal monotone operators are considered. The key results about maximal monotone operators that have been proved over the years are given in Chapter 3. As we have stated earlier, some of these results hold only in reflexive Banach spaces. We will provide some classical counterexamples when some of these results fail in nonreflexive Ba- nach spaces. Here the interest is mostly on the ranges and the domains of maximal monotone operators. In Chapter 4, we will discuss the representations of monotone operators by convex functions, a topic which has generated a lot of interest of late. Here what we are particularly interested in are the Fitzpatrick representations. This is because they are much more simpler and easier to understand than the other available representations. We will discuss the fundamental prop- erties of these representations and we will also demonstrate their usefulness by applying them to obtain a classical result on monotone operators. The Rockafellar sum theorem which is one of the most important results on maximal monotone operators will be discussed in Chapter 5. The most outstanding problem in this theory is whether the Rockafellar sum theorem is valid in nonreflexive Banach spaces. Chapter 5 also discusses several im- portant generalizations of the Rockafellar sum theorem. Some very recent generalizations due to S. Simons - C. Zalinescu are also discussed here. Here we only discuss the reflexive case and a discussion of the nonreflexive case is given in the final Chapter once we have developed enough tools to handle it. We will introduce the subclasses of maximal monotone operator, a very im- portant concept for the study of the nonreflexive case, in Chapter 6. The motivation behind the introduction of these subclasses was to extend some of the known results on reflexive Banach spaces to nonreflexive Banach spaces. Here we will discuss the properties of some of the major subclasses and the vii known relationships among them. Finally as mentioned before, we will conclude with a discussion of the Rock- afellar sum theorem in nonreflexive Banach spaces. Here we will collect most of the special cases where the sum theorem is known to hold in the nonreflex- ive Banach spaces. Obviously, these special cases are in no way a guarantee that the sum theorem holds in general in nonreflexive Banach spaces. But they are nevertheless useful as they give us some clues as to the places where it is improbable to find counterexamples to the sum theorem. As concluding remarks we will include some of the open problems in the theory monotone operators. Chapter 0 Tools In this chapter we collect together the important results from Functional Analysis and Convex Analysis, that we will need in order to describe the The Theory of Monotone Operators.
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