Progress in Nonlinear Differential Equations and Their Applications Volume 75 Editor Haim Brezis Université Pierre et Marie Curie Paris and Rutgers University New Brunswick, N.J. Editorial Board Antonio Ambrosetti, Scuola Internazionale Superiore di Studi Avanzati, Trieste A. Bahri, Rutgers University, New Brunswick Felix Browder, Rutgers University, New Brunswick Luis Cafarelli, Institute for Advanced Study, Princeton Lawrence C. Evans, University of California, Berkeley Mariano Giaquinta, University of Pisa David Kinderlehrer, Carnegie-Mellon University, Pittsburgh Sergiu Klainerman, Princeton University Robert Kohn, New York University P.L. Lions, University of Paris IX Jean Mahwin, Université Catholique de Louvain Louis Nirenberg, New York University Lambertus Peletier, University of Leiden Paul Rabinowitz, University of Wisconsin, Madison John Toland, University of Bath Differential Equations, Chaos and Variational Problems Vasile Staicu Editor Birkhäuser Basel · Boston · Berlin Editor: Vasile Staicu Department of Mathematics University of Aveiro 3810-193 Aveiro Portugal e-mail: [email protected] 2000 Mathematics Subject Classification: 34, 35, 37, 39, 45, 49, 58, 70, 76, 80, 91, 93 Library of Congress Control Number: 2007935492 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at <http://dnb.ddb.de>. ISBN 978-3-7643-8481-4 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8481-4 e-ISBN 978-3-7643-8482-1 9 8 7 6 5 4 3 2 1 www.birkhauser.ch Contents Editorial Introduction ...................................................... ix R.P. Agarwal, M.E. Filippakis, D. O’Regan and N.S. Papageorgiou Nodal and Multiple Constant Sign Solution for Equations with the p-Laplacian ....................................................... 1 Z. Artstein A Young Measures Approach to Averaging ........................ 15 J.-P. Aubin and P. Saint-Pierre Viability Kernels and Capture Basins for Analyzing the Dynamic Behavior: Lorenz Attractors, Julia Sets, and Hutchinson’s Maps ... 29 R. Baier Generalized Steiner Selections Applied to Standard Problems of Set-Valued Numerical Analysis .................................... 49 S. Bianchini On the Euler-Lagrange Equation for a Variational Problem ........ 61 A. Bressan Singular Limits for Impulsive Lagrangian Systems with Dissipative Sources ........................................................... 79 P. Cannarsa, H. Frankowska and E.M. Marchini Lipschitz Continuity of Optimal Trajectories in Deterministic Optimal Control .................................................. 105 C. Carlota and A.Ornelas An Overview on Existence of Vector Minimizers for Almost Convex 1−dim Integrals ................................................... 117 A. Cellina Strict Convexity, Comparison Results and Existence of Solutions to Variational Problems .............................................. 123 vi Contents A. Cernea Necessary Optimality Conditions for Discrete Delay Inclusions .... 135 F. Clarke Necessary Conditions in Optimal Control and in the Calculus of Variations ......................................................... 143 C. Corduneanu Almost Periodicity in Functional Equations ....................... 157 A. Dawidowicz and A. Poskrobko Age-dependent Population Dynamics with the Delayed Argument . 165 R. Dila˜o and R. Alves-Pires Chaos in the Sto¨rmer Problem .................................... 175 F. Ferreira, A.A. Pinto, and D.A. Rand Hausdorff Dimension versus Smoothness .......................... 195 A. Gavioli and L. Sanchez On Bounded Trajectories for Some Non-Autonomous Systems ..... 211 E. Girejko and Z. Bartosiewicz On Generalized Differential Quotients and Viability ............... 223 R. Gon¸calves, A.A. Pinto, and F. Calheiros Nonlinear Prediction in Riverflow – the Paiva River Case .......... 231 J. Kennedy and J.A. Yorke Shadowing in Higher Dimensions .................................. 241 J. Mawhin Boundary Value Problems for Nonlinear Perturbations of Singular φ-Laplacians ...................................................... 247 F.M. Minho´s Existence, Nonexistence and Multiplicity Results for Some Beam Equations ......................................................... 257 S¸t. Mirica˘ Reducing a Differential Game to a Pair of Optimal Control Problems ......................................................... 269 B.S. Mordukhovich Optimal Control of Nonconvex Differential Inclusions .............. 285 Contents vii F. Mukhamedov and J.F.F. Mendes On Chaos of a Cubic p-adic Dynamical System .................... 305 J. Myjak Some New Concepts of Dimension ................................. 317 R. Ortega Degree and Almost Periodicity .................................... 345 M. Oˆtani L∞-Energy Method, Basic Tools and Usage ....................... 357 B. Ricceri Singular Set of Certain Potential Operators in Hilbert Spaces ..... 377 J.M.R. Sanjurjo Shape and Conley Index of Attractors and Isolated Invariant Sets . 393 M.R. Sidi Ammi and D.F.M. Torres Regularity of Solutions for the Autonomous Integrals of the Calculus of Variations ............................................. 407 S. Terracini Multi-modal Periodic Trajectories in Fermi–Pasta–Ulam Chains ... 415 S. Zambrano and M. A. F. Sanjua´n Control of Transient Chaos Using Safe Sets in Simple Dynamical Systems ........................................................... 425 Editorial Introduction Thisbookisacollectionoforiginalpapersandstate-of-the-artcontributionswrit- ten by leading experts in the areas of differential equations, chaos and variational problems in honour of Arrigo Cellina and James A. Yorke, whose remarkable sci- entificcarrierwasasourceofinspirationtomanymathematicians,ontheoccasion of their 65th birthday. Arrigo Cellina and James A. Yorke were born on the same day: August 3, 1941. Both received their Ph.D. degrees from the University of Maryland, where they met first in the late 1960s, at the Institute for Fluid Dynamics and Applied Mathematics. They had offices next to each other and though they were of the same age, Yorke was already Assistant Professor, while Cellina was a Graduate Student. Each one of them had a small daughter, and this contributed to their friendship. Arrigo Cellina James A. Yorke Yorke arrived at the office every day with a provision of cans of Coca Cola, his daily ration, that he put in the air conditioning fan, to keep cool. Cellina says that he was very impressed by Yorke’s way of doing mathematics; Yorke could prove very interesting new results using almost elementary mathematical tools, little more than second year Calculus. From those years, he remembers for example the article Noncontinuable so- lutions of differential-delay equations where Yorke shows, in an elementary way but with a clever use of the extension theorem, that the basic theorem of continu- ation of solutions to ordinary differential equations cannot be valid for functional x Editorial Introduction equations (at that time very fashionable). In the article A continuous differen- tial equation in Hilbert space without existence, Yorke gave the first example of the nonexistence of solutions to Cauchy problems for an ordinary differential in a Hilbert space. Furthermore, in a joint paper with one of his students, Saperstone, he proved a controllability theorem without using the hypothesis that the origin belongs to the interior of the set of controls. This is just a sample of important problems to which Yorke made nontrivial contributions. Yorke went around always carrying in his pocket a notebook where he anno- tated the mathematical problems that seemed important for future investigation. In those years Yorke’s collaboration with Andrezj Lasota began, which produced outstanding results in the theory of “chaos”. Yorke became famous even in non- mathematical circles for his mathematical model for the spread of gonorrhoea. While traditional models were not in accord with experimental data, he proposed asimplemodelbasedontheexistenceoftwogroupsofpeopleandprovedthatthis model fits well the experimental data. Later, in a 1975 paper entitledPeriod three implies chaos withT.Y.Lee,Yorkeintroducedarigorousmathematical definition oftheterm“chaos”forthestudyofdynamicalsystems.Fromthenon,heplayeda leadingroleinthefurtherresearchonchaos,includingitscontrolandapplications. Yorke’s goals to explore interdisciplinary mathematics were fully realized af- terheearnedhisPh.D.andjoinedthefacultyoftheInstituteforPhysicalScience and Technology (IPST), an institute established in 1950 to foster excellence in interdisciplinary research and education at the University of Maryland. He said: All along the goal of myself and my fellow researchers here at Maryland has been to find the concepts that the applied scientist needs. His chaos research group in- troduced many basic concepts with exotic names like crises, the control of chaos, fractal basin boundary, strange non-chaotic attractors, and the Kaplan–Yorke di- mension. One remarkable application of Yorke’s theory of chaos has been the weather prediction. In2003YorkesharedwithBenoitMandelbrotofYaleUniversitytheprizefor Science and Technology of Complexity of the Science and Technology Foundation of Japan for the Creation of Universal Concepts in Complex Systems-Chaos and Fractals. With this prize, Jim Yorke was recognized for his outstanding achieve- ments in nonlinear dynamics that have greatly advanced the frontiers of science and technology. Yorke’sresearchhasbeenhighlyinfluential,withsomeofhispapersreceiving hundreds of citations. He is the author of three books on chaos, of a monograph ongonorrhoeaepidemiology,andofmorethan300papersintheareasofordinary differentialequations,dynamicalsystems,delaydifferentialequations,appliedand random dynamical systems. He believes that a Ph.D. in mathematics is a licence to investigate the uni- verse, and he has supervised over 40 Ph.D. dissertations in the departments of mathematics, physics and computer science. Editorial Introduction xi Currently, Jim Yorke is a Distinguished University Professor of Mathemat- ics and Physics, and Chair of the Mathematics Department of the University of Maryland. ArrigoCellinareceivedaPh.D.degreeinmathematicsin1968andwentback to Italy, where he was Assistant Professor and then Full Professor at the Univer- sities of Perugia, Florence, and Padua, at the International School for Advanced Studies (SISSA) in Trieste, and at the University of Milan. He was a member of thescientificcommitteeandthenDirector(1999–2001)oftheInternationalMath- ematical Summer Centre (CIME) in Florence, Italy, and also a member of the scientific council of CIM (International Centre for Mathematics) seated in Coim- bra, Portugal. Presently he is Professor at the University of Milan “Bicocca” and coordinator of the Doctoral Program of this university. In Italy, the International School for Advanced Studies (SISSA) was estab- lished in 1978, in Trieste, as a dedicated and autonomous scientific institute to develop top-level research in mathematics, physics, astrophysics, biology and neu- roscience, and to provide qualified graduate training to Italian and foreign laure- ates, to train them for research and academic teaching. SISSA was the first Italian school to set up post-laurea courses aimed at a Ph.D. degree (Doctor Philosophiae). Cellina was one of the professors, founders and, for several years, the Coordinator of the Sector of Functional Analysis and Applications at SISSA, from 1978 until 1996. I was lucky to have been initiated to mathematical research on Aubin– Cellina’s book Differential inclusions in a research seminar at the University of Bucharest.ThreeyearslaterIbeganmyPh.D.studiesondifferentialinclusionsat SISSA, under the supervision of Arrigo Cellina. I arrived at SISSA coming from Florence where I spent a very rewarding and training period of one year as a Re- searchFellowofGNAFAunderthesupervisionofRobertoConti,andIremember that Arrigo welcomed me with a kindness equal to his erudition. Always available to discuss and to help his students to overcome difficulties, not only of mathematical orders, Arrigo taught me a lot more than differential inclusions. I remember with great pleasure his beautiful lessons, the long hours of reflectioninfrontoftheblackboardinhisoffice,aswellasthewalksalongthesea or in the park of Miramare. IrememberSISSAofthosedaysasaveryexcitingenvironment.Acommunity of researchers worked there, while several others were visiting SISSA and gave short courses or seminars concerning their new results. The Sector of Functional Analysis and Applications was located in a beautiful place, close to the Castle of Miramare,andneartheInternationalCentreforTheoreticalPhysics(ICTP),with an excellent library where we could spend much of our time. Without a doubt, this has been a very fruitful and rewarding period of my life, both as a scientific and as a life experience. Cellina’s contribution has been significant. Cellina’s scientific work has always been highly original, introducing entirely new techniques to attack the difficult problems he considered. He introduced the notion of graph approximate selection for upper semicontinuous multifunctions,