טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentSeadia Bartz
SubjectOn the Subdifferential
DepartmentDepartment of Mathematics
Supervisor Professor Emeritus Reich Simeon


Abstract

The thesis unifies the theory of subdifferentials of lower semicontinuous convex functions developed during three major breakthroughs. Chapters 1, 2 and 3 of the thesis are organized to summarize results from each period in which a breakthrough took place. The thesis is, mostly, self-contained. It begins with a study of the first breakthrough, that is, the introduction of Fenchel's conjugate function. The second breakthrough took place in the 60's. Two of the most important results established during that period are Rockafellar's characterization of the subdifferential as a maximal cyclically monotone mapping, and Rockafellar's observation that subdifferentials are maximal monotone. The last breakthrough is four years old. This breakthrough began after more attention was drawn to the Fitzpatrick function and the Fitzpatrick family assigned to any maximal monotone mapping. One new development in the thesis is the introduction of a sequence of higher order Fitzpatrick functions. The limit of this sequence is calculated and its close relationship to Rockafellar's antiderivative is pointed out. This is the first time that cyclical monotonicity of the subdifferential is linked to its Fitzpatrick family and a new connection between the breakthrough of forty years ago and the recent one is established. A new practical application of cyclical monotonicty of the subdifferential is presented. This application is a new proof of the convexity of the set of resolvents of subdifferentials on a real Hilbert space. On the other hand, it is pointed out that the set of firmly nonexpansive mappings on a general Banach space need not be a convex set. Other new results are examples and calculations of Fitzpatrick functions and Fitzpatrick families of subdifferentials of some important convex functions. One of these examples is a generalization of a result originally due to Fitzpatrick and Burachik which asserts that the Fitzpatrick function of the subdifferential of a lower semicontinuous sublinear function is the only member of the corresponding Fitzpatrick family. In this case the sequence of the Fitzpatrick functions of all orders is a constant sequence. A general discussion of Fitzpatrick functions is included. A new example of a lower semicontinuous sublinear function which is not subdifferentiable at a point in its domain is presented. The technique which was used to construct this example gives rise to a new, simple proof of a density theorem originally due to Phelps.