Playing Around Resonance An Invitation to the Search of Periodic Sol
This book provides an up-to-date description of the methods needed to face the existence of solutions to some nonlinear boundary value problems. All important and interesting aspects of the theory of periodic solutions of ordinary differential equations r
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Alessandro Fonda
Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations
Birkhäuser Advanced Texts Basler Lehrbücher
Series editors Steven G. Krantz, Washington University, St. Louis, USA Shrawan Kumar, University of North Carolina at Chapel Hill, Chapel Hill, USA Jan Nekováˇr, Université Pierre et Marie Curie, Paris, France
More information about this series at: http://www.springer.com/series/4842
Alessandro Fonda
Playing Around Resonance An Invitation to the Search of Periodic Solutions for Second Order Ordinary Differential Equations
Alessandro Fonda Dipartimento di Matematica e Geoscienze UniversitJa degli Studi di Trieste Trieste, Italy
ISSN 1019-6242 ISSN 2296-4894 (electronic) BirkhRauser Advanced Texts Basler LehrbRucher ISBN 978-3-319-47089-4 ISBN 978-3-319-47090-0 (eBook) DOI 10.1007/978-3-319-47090-0 Library of Congress Control Number: 2016958441 © Springer International Publishing AG 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This book is published under the trade name Birkhäuser, www.birkhauser-science.com The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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Contents
1
Preliminaries on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 The Hilbert Space Structure . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Some Examples of Hilbert Spaces. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Fundamental Properties . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 Subspaces .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.5 Orthogonal Subspaces.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .
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