HANDBOOK OF DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS VOLUME I This Page Intentionally Left Blank HANDBOOK OF DIFFERENTIAL EQUATIONS ORDINARY DIFFERENTIAL EQUATIONS VOLUME I 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 2004 NORTH HOLLAND Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER B.V. ELSEVIER Inc. Sara Burgerhartstraat 25 525 B Street, Suite 1900 P.O. Box 211, 1000 AE Amsterdam San Diego, CA 92101-4495 The Netherlands USA ELSEVIER Ltd ELSEVIER Ltd The Boulevard, Langford Lane 84 Theobalds Road Kidlington, Oxford OX5 1GB London WC1X 8RR UK UK © 2004 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Pub- lisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier’s Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: Preface Ordinary differential equations is a wide mathematical discipline which is closely related to both pure mathematical research and real world applications. Most mathematical formula- tions of physical laws are described in terms of ordinary and partial differential equations, and this has been a great motivation for their study in the past. In the 20th century the extremely fast development of Science led to applications in the fields of chemistry, bi- ology, medicine, population dynamics, genetic engineering, economy, social sciences and others, as well. All these disciplines promoted to higher level and new discoveries were made with the help of this kind of mathematical modeling. At the same time, real world problems have been and continue to be a great inspiration for pure mathematics, particu- larly concerning ordinary differential equations: they led to new mathematical models and challenged mathematicians to look for new methods to solve them. It should also be mentioned that an extremely fast development of computer sciences took place in the last three decades: mathematicians have been provided with a tool which had not been available before. This fact encouraged scientists to formulate more complex mathematical models which, in the past, could hardly be resolved or even understood. Even if computers rarely permit a rigorous treatment of a problem, they are a very useful tool to get concrete numerical results or to make interesting numerical experiments. In the field of ordinary differential equations this phenomenon led more and more mathematicians to the study of nonlinear differential equations. This fact is reflected pretty well by the contributions to this volume. The aim of the editors was to collect survey papers in the theory of ordinary differential equations showing the “state of the art”, presenting some of the main results and methods to solve various types of problems. The contributors, besides being widely acknowledged experts in the subject, are known for their ability of clearly divulging their subject. We are convinced that papers like the ones in this volume are very useful, both for the experts and particularly for younger research fellows or beginners in the subject. The editors would like to express their deepest gratitude to all contributors to this volume for the effort made in this direction. The contributions to this volume are presented in alphabetical order according to the name of the first author. The paper by Agarwal and O’Regan deals with singular initial and boundary value problems (the nonlinear term may be singular in its dependent variable and is allowed to change sign). Some old and new existence results are established and the proofs are based on fixed point theorems, in particular, Schauder’s fixed point theo- rem and a Leray–Schauder alternative. The paper by De Coster and Habets is dedicated to the method of upper and lower solutions for boundary value problems. The second order equations with various kinds of boundary conditions are considered. The emphasis is put v vi Preface on well ordered and non-well ordered pairs of upper and lower solutions, connection to the topological degree and multiplicity of the solutions. The contribution of Došlý deals with half-linear equations of the second order. The principal part of these equations is rep- resented by the one-dimensional p-Laplacian and the author concentrates mainly on the oscillatory theory. The paper by Jacobsen and Schmitt is devoted to the study of radial solutions for quasilinear elliptic differential equations. The p-Laplacian serves again as a prototype of the main part in the equation and the domains as a ball, an annual region, the exterior of a ball, or the entire space are under investigation. The paper by Llibre is dedicated to differential systems or vector fields defined on the real or complex plane. The author presents a deep and complete study of the existence of first integrals for planar poly- nomial vector fields through the Darbouxian theory of integrability. The paper by Mawhin takes the simple forced pendulum equation as a model for describing a variety of nonlinear phenomena: multiplicity of periodic solutions, subharmonics, almost periodic solutions, stability, boundedness, Mather sets, KAM theory and chaotic dynamics. It is a review pa- per taking into account more than a hundred research articles appeared on this subject. The paper by Srzednicki is a review of the main results obtained by the Waz˙ewski method in the theory of ordinary differential equations and inclusions, and retarded functional dif- ferential equations, with some applications to boundary value problems and detection of chaotic dynamics. It is concluded by an introduction of the Conley index with examples of possible applications. Last, but not least, we thank the Editors at Elsevier, who gave us the opportunity of making available a collection of articles that we hope will be useful to mathematicians and scientists interested in the recent results and methods in the theory and applications of ordinary differential equations. List of Contributors Agarwal, R.P., Florida Institute of Technology, Melbourne, FL (Ch. 1) De Coster, C., Université du Littoral, Calais Cédex, France (Ch. 2) Došlý, O., Masaryk University, Brno, Czech Republic (Ch. 3) Habets, P., Université Catholique de Louvain, Louvain-la-Neuve, Belgium (Ch. 2) Jacobsen, J., Harvey Mudd College, Claremont, CA (Ch. 4) Llibre, J., Universitat Autónoma de Barcelona, Bellaterra, Barcelona, Spain (Ch. 5) Mawhin, J., Université Catholique de Louvain, Louvain-la-Neuve, Belgium (Ch. 6) O’Regan, D., National University of Ireland, Galway, Ireland (Ch. 1) Schmitt, K., University of Utah, Salt Lake City, UT (Ch. 4) Srzednicki, R., Institute of Mathematics, Jagiellonian University, Kraków, Poland (Ch. 7) vii This Page Intentionally Left Blank Contents Preface v List of Contributors vii 1. A survey of recent results for initial and boundary value problems singular in the dependent variable 1 R.P. Agarwal and D. O’Regan 2. The lower and upper solutions method for boundary value problems 69 C. De Coster and P. Habets 3. Half-linear differential equations 161 O. Došlý 4. Radial solutions of quasilinear elliptic differential equations 359 J. Jacobsen and K. Schmitt 5. Integrability of polynomial differential systems 437 J. Llibre 6. Global results for the forced pendulum equation 533 J. Mawhin 7. Waz˙ewski method and Conley index 591 R. Srzednicki Author Index 685 Subject Index 693 ix