Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems

Frontmatter

Chapter I. Subdifferentiability and Duality Mappings

Abstract

Our aim in this Chapter is to introduce the notion of duality mapping on a Banach space X; to this purpose we survey basic results of convex analysis and connect them with the differentiability properties of the norm on X.

Ioana Cioranescu

Chapter II. Characterizations of Some Classes of Banach Spaces by Duality Mappings

Abstract

In this chapter we shall characterize some classes of Banach spaces, among which strictly convex spaces, uniformly convex spaces and reflexive Banach spaces in terms of properties of the duality mapping such as continuity, injectivity or surjectivity. Some applications to L^p and 1^p spaces are given.

Ioana Cioranescu

Chapter III. Renorming of Banach Spaces

Abstract

This chapter is mainly concerned with the basic renorming theorems: the renorminy of c₀(Γ) and the theorems of Lindenstrauss and Trojanski. Moreover Aspund’s averaging techinque is presented.

Ioana Cioranescu

Chapter IV. On the Topological Degree in Finite and Infinite Dimensions

Abstract

We give a short (analytical) presentation of Brouwer’s theory of the topological degree of continuous mappings in finite dimensional Banach spaces and a generalization of it to A proper mappings in Banach spaces. The connection with the Leray-Schauder degree is commented and some applications to the topological degree of normalized duality mappings are made.

Ioana Cioranescu

Chapter V. Nonlinear Monotone Mappings

Abstract

In this chapter we shall present various results on nonlinear monotone mappings in Banach spaces, pointing out further properties of duality mappings. Applications are made to some nonlinear functional equations.

Ioana Cioranescu

Chapter VI. Accretive Mappings and Semigroups of Nonlinear Contractions

Abstract

In this chapter we first present general result on accretive mappings on a Banach space X in order to tackle then the problem of the genration of semigroups of nonlinear contractions.

Ioana Cioranescu

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