Geometric Properties of Banach Spaces and Nonlinear Iterations
Nonlinear functional analysis and applications is an area of study that has provided fascination for many mathematicians across the world. This monograph delves specifically into the topic of the geometric properties of Banach spaces and nonlinear iterati
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1965
Charles Chidume
Geometric Properties of Banach Spaces and Nonlinear Iterations
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Charles Chidume Abdus Salam International Centre for Theoretical Physics Mathematics Section Strada Costiera 11 34014 Trieste Italy [email protected]
ISBN: 978-1-84882-189-7 e-ISBN: 978-1-84882-190-3 DOI: 10.1007/978-1-84882-190-3 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008938194 Mathematics Subject Classification (2000): 47XX, 46XX, 45XX, 49XX, 65XX, 68XX c Springer-Verlag London Limited 2009 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. 987654321 Springer Science+Business Media, LLC springer.com
To my beloved family: Ifeoma (wife), and children: Chu Chu; Ada; KK and Okey (Okido).
Preface
The contents of this monograph fall within the general area of nonlinear functional analysis and applications. We focus on an important topic within this area: geometric properties of Banach spaces and nonlinear iterations, a topic of intensive research efforts, especially within the past 30 years, or so. In this theory, some geometric properties of Banach spaces play a crucial role. In the first part of the monograph, we expose these geometric properties most of which are well known. As is well known, among all infinite dimensional Banach spaces, Hilbert spaces have the nicest geometric properties. The availability of the inner product, the fact that the proximity map or nearest point map of a real Hilbert space H onto a closed convex subset K of H is Lipschitzian with constant 1, and the following two identities ||x + y||2 = ||x||2 + 2x, y + ||y||2 , ||λx + (1 − λ)y||2 = λ||x||2 + (1 − λ)||y||2 − λ(1 − λ)||x − y||2 ,
(∗) (∗∗)
which hold for all x, y ∈ H, are some of the geometric properties that characterize inner product spaces and also make certain problems posed in Hilbert spaces more manageable than those in general Banach spaces. However, as has been rightly observed by M. Hazewinkel, “... many, and probably most, mathematical objects and models do not naturally live in Hilbert spaces”. Consequently, to extend s
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