Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems

ISBN-13: 9789401074544 / Angielski / Miękka / 2012 / 260 str.

I. Cioranescu

Opis

One service mathematics has rendered the 'Et moi, ...* si Javait so comment en revenir. je n'y serais point alle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- The series is divergent; therefore we may be sense'. Eric T. Bell able to do something with it. o. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics ...'; 'One service logic has rendered com- puter science ...'; 'One service category theory has rendered mathematics ...'. AIl arguably true. And all statements obtainable this way form part of the raison d'etre of this series.

Spis treści

I. Subdifferentiability and Duality Mappings.- § 1. Generalities on convex functions.- § 2. The subdifferential and the conjugate of a convex function.- § 3. Smooth Banach spaces.- § 4. Duality mappings on Banach spaces.- § 5. Positive duality mappings.- Exercises.- Bibliographical comments.- II Characterizations of Some Classes of Banach Spaces by Duality Mappings.- § 1. Strictly convex Banach spaces.- § 2. Uniformly convex Banach spaces.- § 3. Duality mappings in reflexive Banach spaces.- § 4. Duality mappings in LP-spaces.- § 5. Duality mappings in Banach spaces with the property (h) and (?)1.- Exercises.- Bibliographical comments.- III Renorming of Banach Spaces.- § 1. Classical renorming results.- § 2. Lindenstrauss’ and Trojanski’s Theorems.- Exercises.- Bibliographical comments.- IV On the Topological Degree in Finite and Infinite Dimensions.- § 1. Brouwer’s degree.- § 2. Browder-Petryshyn’s degree for A-proper mappings.- § 3. P-compact mappings.- Exercises.- Bibliographical comments.- V Nonlinear Monotone Mappings.- § 1. Demicontinuity and hemicontinuity for monotone operators.- § 2. Monotone and maximal monotone mappings.- § 3. The role of the duality mapping in surjectivity and maximality problems.- § 4. Again on subdifferentials of convex functions.- Exercises.- Bibliographical comments.- VI Accretive Mappings and Semigroups of Nonlinear Contractions.- § 1. General properties of maximal accretive mappings.- § 2. Semigroups of nonlinear contractions in uniformly convex Banach spaces.- § 3. The exponential formula of Crandall-Liggett.- § 4. The abstract Cauchy problem for accretive mappings.- § 5. Semigroups of nonlinear contractions in Hilbert spaces.- § 6. The inhomogeneous case.- Exercises.- Bibliographical comments.- References.