Canada a drabek p fonda a (eds ) handbook of differential equations ordinary differential equations vol 3

H ANDBOOK OF D IFFERENTIAL E QUATIONS O RDINARY D IFFERENTIAL E QUATIONS VOLUME III This page intentionally left blank H ANDBOOK OF D IFFERENTIAL E QUATIONS O RDINARY D IFFERENTIAL E QUATIONS VOLUME III Edited by A CAÑADA Department of Mathematical Analysis, Faculty of Sciences, University of Granada, Granada, Spain P DRÁBEK Department of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Pilsen, Czech Republic A FONDA Department of Mathematical Sciences, Faculty of Sciences, University of Trieste, Trieste, Italy Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2006 Copyright © 2006 Elsevier B.V All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: permissions@elsevier.com Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/ locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN-13: 978-0-444-52849-0 ISBN-10: 0-444-52849-0 Set ISBN: 444 51742-1 For information on all North-Holland publications visit our website at books.elsevier.com Printed and bound in The Netherlands 06 07 08 09 10 10 Preface This is the third volume in a series devoted to self contained and up-to-date surveys in the theory of ordinary differential equations, written by leading researchers in the area All contributors have made an additional effort to achieve readability for mathematicians and scientists from other related fields, in order to make the chapters of the volume accessible to a wide audience These ideas faithfully reflect the spirit of this multi-volume and the editors hope that it will become very useful for research, learning and teaching We express our deepest gratitude to all contributors to this volume for their clearly written and elegant articles This volume consists of seven chapters covering a variety of problems in ordinary differential equations Both, pure mathematical research and real word applications are reflected pretty well by the contributions to this volume They are presented in alphabetical order according to the name of the first author The paper by Andres provides a comprehensive survey on topological methods based on topological index, Lefschetz and Nielsen numbers Both single and multivalued cases are investigated Ordinary differential equations are studied both on finite and infinite dimensions, and also on compact and noncompact intervals There are derived existence and multiplicity results Topological structures of solution sets are investigated as well The paper by Bonheure and Sanchez is dedicated to show how variational methods have been used in the last 20 years to prove existence of heteroclinic orbits for second and fourth order differential equations having a variational structure It is divided in parts: the first one deals with second order equations and systems, while the second one describes recent results on fourth order equations The contribution by De Coster, Obersnel and Omari deals with qualitative properties of solutions of two kinds of scalar differential equations: first order ODEs, and second order parabolic PDEs Their setting is very general, so that neither uniqueness for the initial value problems nor comparison principles are guaranteed They particularly concentrate on periodic solutions, their localization and possible stability The paper by Han is dedicated to the theory of limit cycles of planar differential systems and their bifurcations It is structured in three main parts: general properties of limit cycles, Hopf bifurcations and perturbations of Hamiltonian systems Many results are closely related to the second part of Hilbert’s 16th problem which concerns with the number and location of limit cycles of a planar polynomial vector field of degree n posed in 1901 by Hilbert The survey by Hartung, Krisztin, Walther and Wu reports about the more recent work on state-dependent delayed functional differential equations These equations appear in a natural way in the modelling of evolution processes in very different fields: physics, automatic control, neural networks, infectious diseases, population growth, cell biology, epidemiology, etc The authors emphasize on particular models and on the emerging theory from the dynamical systems point v vi Preface of view The paper by Korman is devoted to two point nonlinear boundary value problems depending on a parameter λ The main question is the precise number of solutions of the problem and how these solutions change with the parameter To study the problem, the author uses bifurcation theory based on the implicit function theorem (in Banach spaces) and on a well known theorem by Crandall and Rabinowitz Other topics he discusses involve pitchfork bifurcation and symmetry breaking, sign changing solutions, etc Finally, the paper by Rach˚unková, Stanˇek and Tvrdý is a survey on the solvability of various nonlinear singular boundary value problems for ordinary differential equations on the compact interval The nonlinearities in differential equations may be singular both in the time and space variables Location of all singular points need not be known With this volume we end our contribution as editors of the Handbook of Differential Equations We thank the staff at Elsevier for efficient collaboration during the last three years List of Contributors Andres, J., Palacký University, Olomouc-Hejˇcín, Czech Republic (Ch 1) Bonheure, D., Université Catholique de Louvain, Louvain-La-Neuve, Belgium (Ch 2) De Coster, C., Université du Littoral-Côte d’Opale, Calais Cédex, France (Ch 3) Han, M., Shanghai Normal University, Shanghai, China (Ch 4) Hartung, F., University of Veszprém, Veszprém, Hungary (Ch 5) Korman, P., University of Cincinnati, Cincinnati, OH, USA (Ch 6) Krisztin, T., University of Szeged, Szeged, Hungary (Ch 5) Obersnel, F., Università degli Studi di Trieste, Trieste, Italy (Ch 3) Omari, P., Università degli Studi di Trieste, Trieste, Italy (Ch 3) Rach˚unková, I., Palacký University, Olomouc, Czech Republic (Ch 7) Sanchez, L., Universidade de Lisboa, Lisboa, Portugal (Ch 2) Stanˇek, S., Palacký University, Olomouc, Czech Republic (Ch 7) Tvrdý, M., Mathematical Institute, Academy of Sciences of the Czech Republic, Praha, Czech Republic (Ch 7) Walther, H.-O., Universität Gießen, Gießen, Germany (Ch 5) Wu, J., York University, Toronto, Canada (Ch 5) vii This page intentionally left blank Contents Preface List of Contributors Contents of Volume Contents of Volume v vii xi xiii Topological principles for ordinary differential equations J Andres Heteroclinic orbits for some classes of second and fourth order differential equations D Bonheure and L Sanchez A qualitative analysis, via lower and upper solutions, of first order periodic evolutionary equations with lack of uniqueness C De Coster, F Obersnel and P Omari Bifurcation theory of limit cycles of planar systems M Han Functional differential equations with state-dependent delays: Theory and applications F Hartung, T Krisztin, H.-O Walther and J Wu Global solution branches and exact multiplicity of solutions for two point boundary value problems P Korman Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations I Rach˚unková, S Stanˇek and M Tvrdý Author index Subject index 103 203 341 435 547 607 725 735 ix ... Edited by A CAÑADA Department of Mathematical Analysis, Faculty of Sciences, University of Granada, Granada, Spain P DRÁBEK Department of Mathematics, Faculty of Applied Sciences, University of West... can be appropriately generalized (see [ 73, Proposition 7.9 on p 229 and Proposition 7. 23 on pp 234 – 235 ]) If ϕ : X Y is only u-Carathéodory and X, Y are (not necessarily separable) Banach spaces,... that (i) X is a (metric) ANR-space, e.g., a retract of an open subset of a convex set in a Fréchet space, (ii) ϕ is a compact (i.e ϕ(X) is compact) composition of an Rδ -map p −1 : X and a continuous