H ANDBOOK OF D IFFERENTIAL E QUATIONS O RDINARY D IFFERENTIAL E QUATIONS VOLUME II This page intentionally left blank H ANDBOOK OF D IFFERENTIAL E QUATIONS O RDINARY D IFFERENTIAL E QUATIONS VOLUME II 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 2005 NORTH HOLLAND Amsterdam • Boston • Heidelberg • London • New York • Oxford • Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo ELSEVIER B.V Radarweg 29 P.O Box 211, 1000 AE Amsterdam The Netherlands ELSEVIER Inc 525 B Street, Suite 1900 San Diego, CA 92101-4495 USA ELSEVIER Ltd The Boulevard, Langford Lane Kidlington, Oxford OX5 1GB UK ELSEVIER Ltd 84 Theobalds Road London WC1X 8RR UK © 2005 Elsevier B.V All rights reserved This work is 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permission of the Publisher Address permissions requests to: Elsevier’s Rights Department, at the fax and e-mail addresses noted above 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 First edition 2005 Library of Congress Cataloging in Publication Data: A catalog record is available from the Library of Congress British Library Cataloguing in Publication Data: A catalogue record is available from the British Library ISBN 444 52027 Set ISBN: 444 51742 ∞ The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper) Printed in The Netherlands Preface This handbook is the second 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 multivolume 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 six 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 Barbu and Lefter is dedicated to the discussion of the first order necessary and sufficient conditions of optimality in control problems governed by ordinary differential systems The authors provide a complete analysis of the Pontriaghin maximum principle and dynamic programming equation The paper by Bartsch and Szulkin is a survey on the most recent advances in the search of periodic and homoclinic solutions for Hamiltonian systems by the use of variational methods After developing some basic principles of critical point theory, the authors consider a variety of situations where periodic solutions appear, and they show how to detect homoclinic solutions, including the so-called “multibump” solutions, as well The contribution of Cârj˘a and Vrabie deals with differential equations on closed sets After some preliminaries on Brezis–Browder ordering principle and Clarke’s tangent cone, the authors concentrate on problems of viability and problems of invariance Moreover, the case of Carathéodory solutions and differential inclusions are considered The paper by Hirsch and Smith is dedicated to the theory of monotone dynamical systems which occur in many biological, chemical, physical and economic models The authors give a unified presentation and a broad range of the applicability of this theory like differential equations with delay, second order quasilinear parabolic problems, etc The paper by López-Gómez analyzes the dynamics of the positive solutions of a general class of planar periodic systems, including those of Lotka– Volterra type and a more general class of models simulating symbiotic interactions within global competitive environments The mathematical analysis is focused on the study of coexistence states and the problem of ascertaining the structure, multiplicity and stability of these coexistence states in purely symbiotic and competitive environments Finally, the paper by Ntouyas is a survey on nonlocal initial and boundary value problems Here, some old and new results are established and the author shows how the nonlocal initial or boundv vi Preface ary conditions generalize the classical ones, having many applications in physics and other areas of applied mathematics We thank again the Editors at Elsevier for efficient collaboration List of Contributors Barbu, V., “Al.I Cuza” University, Ia¸si, Romania, and “Octav Mayer” Institute of Mathematics, Romanian Academy, Ia¸si, Romania (Ch 1) Bartsch, T., Universität Giessen, Giessen, Germany (Ch 2) Cârj˘a, O., “Al I Cuza” University, Ia¸si, Romania (Ch 3) Hirsch, M.W., University of California, Berkeley, CA (Ch 4) Lefter, C., “Al.I Cuza” University, Ia¸si, Romania, and “Octav Mayer” Institute of Mathematics, Romanian Academy, Ia¸si, Romania (Ch 1) López-Gómez, J., Universidad Complutense de Madrid, Madrid, Spain (Ch 5) Ntouyas, S.K., University of Ioannina, Ioannina, Greece (Ch 6) Smith, H., Arizona State University, Tempe, AZ (Ch 4) Szulkin, A., Stockholm University, Stockholm, Sweden (Ch 2) Vrabie, I.I., “Al I Cuza” University, Ia¸si, Romania, and “Octav Mayer” Institute of Mathematics, Romanian Academy, Ia¸si, Romania (Ch 3) vii This page intentionally left blank Contents Preface List of Contributors Contents of Volume v vii xi Optimal control of ordinary differential equations V Barbu and C Lefter Hamiltonian systems: periodic and homoclinic solutions by variational methods T Bartsch and A Szulkin Differential equations on closed sets O Cârj˘a and I.I Vrabie Monotone dynamical systems M.W Hirsch and H Smith Planar periodic systems of population dynamics J López-Gómez Nonlocal initial and boundary value problems: a survey S.K Ntouyas Author index Subject index 77 147 239 359 461 559 565 ix 556 S.K Ntouyas [48] G.L Karakostas and P.Ch Tsamatos, Positive solutions for a nonlocal boundary-value problem with increasing response, Electron J Differential Equations 73 (2000), pp (electronic) [49] G.L Karakostas and P.Ch Tsamatos, Positive solutions of a boundary-value problem for second order ordinary differential equations, Electron J Differential Equations 49 (2000), pp (electronic) [50] G.L Karakostas and P.Ch Tsamatos, On a nonlocal boundary value problem at resonance, J Math Anal Appl 259 (2001), 209–218 [51] G.L Karakostas and P.Ch Tsamatos, Multiple positive solutions for a nonlocal boundary-value problem with response function quiet at zero, Electron J Differential Equations 13 (2001), 10 pp (electronic) [52] G.L Karakostas and P.Ch Tsamatos, Sufficient conditions for the existence of nonnegative solutions of a nonlocal boundary value problem, Appl Math Lett 15 (2002), 401–407 [53] G.L Karakostas and P.Ch Tsamatos, Nonlocal boundary vector value problems for ordinary differential systems of higher order, Nonlinear Anal 51 (2002), 1421–1427 [54] M Krasnosel’skii, Positive Solutions of Operator Equations, Noordhoff, Groningen (1964) [55] R Legget and L Williams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ Math J 28 (1979), 673–688 [56] B Liu, Positive solutions of a nonlinear three-point boundary value problem, Appl Math Comput 132 (2002), 11–28 [57] B Liu, Positive solutions of a nonlinear three-point boundary value problem, Comput Math Appl 44 (2002), 201–211 [58] B Liu, Positive solutions of a nonlinear three-point boundary value problem, Appl Math Comput 132 (2002), 11–28 [59] B Liu, Solvability of multi-point boundary value problem at resonance IV, Appl Math Comput 143 (2003), 275–299 [60] B Liu, Solvability of multi-point boundary value problem at resonance II, Appl Math Comput 136 (2003), 353–377 [61] B Liu and J Yu, Solvability of multi-point boundary value problems at resonance I, Indian J Pure Appl Math 33 (2002), 475–494 [62] B Liu and J Yu, Solvability of multi-point boundary value problem at resonance III, Appl Math Comput 129 (2002), 119–143 [63] X Liu, Nontrivial solutions of singular m-point boundary value problems, J Math Anal Appl 284 (2003), 576–590 [64] X Liu, J Qiu and Y Guo, Three positive solutions for second-order m-point boundary value problems, Appl Math Comput 156 (2004), 733–742 [65] Y Liu and W Ge, Positive solutions for (n − 1, 1) three-point boundary value problems with coefficient that changes sign, J Math Anal Appl 282 (2003), 816–825 [66] Y Liu and W Ge, Multiple positive solutions to a three-point boundary value problem with p-Laplacian, J Math Anal Appl 277 (2003), 293–302 [67] R Ma, Existence theorems for a second order m-point boundary value problem, J Math Anal Appl 211 (1997), 545–555 [68] R Ma, Existence theorems for a second order three-point boundary value problem, J Math Anal Appl 212 (1997), 430–442 [69] R Ma, Positive solutions of a nonlinear three-point boundary value problem, Electron J Differential Equations 34 (1999), pp (electronic) [70] R Ma, Multiplicity of positive solutions for second-order three-point boundary value problems, Comput Math Appl 40 (2000), 193–204 [71] R Ma, Existence of solutions of a nonlinear m-point boundary value problems, J Math Anal Appl 256 (2001), 556–567 [72] R Ma, Positive solutions of a nonlinear m-point boundary value problem, Comput Math Appl 42 (2001), 755–765 [73] R Ma, Positive solutions for second order functional differential equations, Dynam Systems Appl 10 (2001), 215–223 [74] R Ma, Existence and uniqueness of solutions to first-order three-point boundary value problems, Appl Math Lett 15 (2002), 211–216 Nonlocal initial and boundary value problems 557 [75] R Ma, Existence of positive solutions for superlinear semipositone m-point boundary value problems, Proc Edinburgh Math Soc 46 (2003), 279–292 [76] R Ma, Multiplicity results for a three-point boundary value problem at resonance, Nonlinear Anal 53 (2003), 777–789 [77] R Ma, Multiple positive solutions for nonlinear m-point boundary value problems, Appl Math Comput 148 (2004), 249–262 [78] R Ma and N Castaneda, Positive solutions for second-order three-point boundary value problems, Appl Math Lett 14 (2001), 1–5 [79] R Ma and H Wang, Positive solutions of nonlinear three-point boundary-value problems, J Math Anal Appl 279 (2003), 216–227 [80] S Marano, A remark on a second order three-point boundary value problem, J Math Anal Appl 183 (1994), 518–522 [81] S Marano, A remark on a third-order three-point boundary value problem, Bull Austral Math Soc 49 (1994), 1–5 [82] J Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, NSF-CBMS Reg Conf Ser Math., Vol 40, Amer Math Soc., Providence, RI (1979) [83] M Moshinsky, Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas, Bol Soc Mat Mexicana (1950), 1–25 [84] K Murty and S Sivasundaram, Existence and uniqueness of solutions to three-point boundary value problems associated with nonlinear first order systems of differential equations, J Math Anal Appl 173 (1993), 158–164 [85] X Ni and W Ge, Multi-point boundary value problems for the p-Laplacian at resonance, Electron J Differential Equations 112 (2003), pp (electronic) [86] S.K Ntouyas and P.Ch Tsamatos, Global existence for semilinear evolution equations with nonlocal conditions, J Math Anal Appl 210 (1997), 679–687 [87] P.K Palamides, Positive and monotone solutions of an m-point boundary value problem, Electron J Differential Equations 18 (2002), 1–16 pp (electronic) [88] B Przeradski and R Stanczy, Solvability of a multi-point boundary value problem at resonance, J Math Anal Appl 264 (2001), 253–261 [89] Y Raffoul, Positive solutions of three-point nonlinear second order boundary value problem, Electron J Qual Theory Differ Equ 15 (2002), 11 pp (electronic) [90] S Timoshenko, Theory of Elastic Stability, McGraw-Hill, New York (1961) [91] J Wang and D Zheng, On the existence of positive solutions to a three-point boundary value problem for the one-dimensional p-Laplacian, Z Angew Math Mech 77 (1997), 477–479 [92] J.R Webb, Positive solutions of some three-point boundary value problems via fixed point index theory, Nonlinear Anal 47 (2001), 4319–4322, [93] M Zhang and Y Han, On the applications of Leray–Schauder continuation theorem to boundary value problems of semilinear differential equations, Ann Differential Equations 13 (1997), 189–207 [94] Z Zhang and J Wang, The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems, J Comput Appl Math 147 (2002), 41–52 [95] Z Zhang and J Wang, Positive solutions to a second order three-point boundary value problem, J Math Anal Appl 285 (2003), 237–249 This page intentionally left blank Author Index Roman numbers refer to pages on which the author (or his/her work) is mentioned Italic numbers refer to reference pages Numbers between brackets are the reference numbers No distinction is made between first and co-author(s) Abbondandolo, A 80, 106, 121, 142 [1] Adams, R.A 98, 126, 129, 142 [2] Aftabizadeh, A 489, 554 [1] Agrachev, A.A 3, 16, 18, 57, 67, 74 [1]; 74 [2] Ahmad, S 439, 459 [1]; 459 [2] Albrecht, F 297, 348 [1] Alexandroff, A 315, 348 [2] Alikakos, N 314, 321, 332, 348 [3] Alvárez, C 439, 459 [3] Amann, H 104, 106, 121, 142 [3]; 142 [4]; 275, 314, 315, 320, 332, 338, 340, 345, 348 [4]; 349 [5]; 349 [6]; 349 [7]; 349 [8]; 349 [9]; 349 [10]; 349 [11]; 398, 459 [4] Ambrosetti, A 80, 142 [5] Amine, Z 458, 459 [5] Angenent, S 349 [12]; 349 [27] Arioli, G 90, 129, 138, 142 [6] Arnold, V.I 5, 6, 8, 19, 27, 74 [3] Aubin, J.P 154, 188, 228, 229, 233, 234, 235 [1]; 235 [2]; 235 [3]; 235 [4] Avery, R 548, 553, 554 [2]; 554 [3] Berestycki, H 91, 115, 121, 142 [7]; 142 [8]; 143 [20] Berliocchi, H 190, 235 [10] Berman, A 282, 284, 286–289, 349 [18]; 349 [19] Birkhoff, G 315, 349 [20] Bitsadze, A 470, 554 [7]; 554 [8] Bliss, G.A 4, 74 [8] Bolotin, S.V 139, 143 [21] Boltyanskii, V.G 29, 75 [30]; 223, 235 [11]; 235 [12]; 237 [83] Bony, J.M 153, 235 [13] Bothe, D 233, 234, 235 [14]; 235 [15] Boucherif, A 482, 484, 554 [9] Bouguima, S 482, 484, 554 [9] Bouligand, H 159, 160, 235 [16] Bressan, A 233, 235 [17] Brezis, H 11, 13, 74 [9]; 153, 154, 156, 157, 235 [18]; 235 [19] Browder, F 156, 157, 235 [19] Brunovsky, P 296, 314, 349 [21]; 349 [22] Byszewski, L 465, 469, 554 [10]; 554 [11]; 554 [12] Bahri, A 91, 142 [7]; 142 [8] Bai, C 546, 554, 554 [4]; 554 [5]; 554 [6] Barbu, V 3, 11, 38, 47, 67, 70, 74 [4]; 74 [5]; 74 [6]; 230, 235 [5] Bartsch, T 88–90, 97, 107, 125, 142 [9]; 142 [10]; 142 [11]; 143 [12]; 143 [13]; 143 [14]; 143 [15]; 143 [16] Batty, C 349 [13] Bebernes, W 227, 235 [6] Bellman, R 3, 74 [7]; 223, 235 [7]; 235 [8]; 235 [9] Benaïm, M 296, 302, 349 [14]; 349 [15]; 349 [16]; 349 [17] Benci, V 80, 88, 97, 106, 121, 143 [17]; 143 [18]; 143 [19] Calvert, B 542, 554 [13] Carathéodory, C 223, 235 [29] Cârj˘a, O 154, 158, 176, 178, 190, 203, 210, 213, 225, 227, 228, 231, 233, 234, 235 [20]; 235 [21]; 235 [22]; 235 [23]; 235 [24]; 235 [25]; 235 [26]; 235 [27]; 235 [28] Castaneda, N 533, 557 [78] Cellina, A 154, 188, 228, 229, 233, 234, 235 [3] Cerami, G 96, 143 [22] Cesari, L 223, 236 [30] Chang, K.C 81, 82, 106, 110, 121, 124, 143 [23] Chen, M 314, 349 [24] Chen, X 314, 349 [24] Chen, X.-Y 314, 349 [23] 559 560 Author Index Chi¸s-Ster, ¸ I 232, 236 [31] Cholewa, J 332, 345, 349 [25] Chow, S.N 331, 349 [26] Clarke, F.H 3, 14, 15, 32, 50, 73, 74 [10]; 74 [11]; 74 [12]; 74 [13]; 74 [14]; 163, 175, 223, 226, 236 [32]; 236 [33]; 236 [34]; 236 [35] Clements, Ph 349 [27] Coddington, E.A 16, 74 [15]; 349 [28] Cohen, D.S 314, 352 [95] Colombo, F 340, 349 [29] Conley, C 124, 143 [24]; 293, 349 [30]; 349 [31] Coppel, W 349 [32] Cornet, B 166, 236 [36] Cosner, C 345, 349 [33] Coti Zelati, V 80, 126, 135, 136, 138–140, 142 [5]; 143 [25]; 143 [26] Courant, R 7, 8, 74 [16]; 223, 236 [37]; 314, 349 [34] Crandall, M.G 68, 70, 74 [17]; 75 [18]; 75 [19]; 75 [20]; 151, 152, 154, 233, 236 [38]; 236 [39]; 371, 404, 407, 459 [6] Cushing, J.M 392, 439, 459 [7] Fang, J 546, 554, 554 [4]; 554 [5]; 554 [6] Federer, H 161, 236 [41] Feferman, S 156, 236 [42] Felmer, P 118, 124, 143 [37]; 143 [38]; 144 [39] Feng, W 497, 505, 554 [15]; 554 [16] Fife, P 350 [49]; 350 [50] Filippov, A.F 213, 236 [43] Fleckinger-Pellé, J 327, 350 [45]; 350 [46] Fleming, W.H 3, 73, 75 [22] Foias, C 350 [43] Fomin, S.V 4, 8, 75 [24] Fonda, A 124, 144 [40] Fournier, G 92, 94, 124, 144 [42] Frankowska, H 190, 228, 234, 235 [4]; 236 [44]; 236 [45] Freedman, H.I 320, 313, 356 [235]; 357 [236] Friedman, A 350 [47]; 350 [48] Frobenius, G 315, 350 [51] Fryszkowski, A 233, 236 [46] Fukaya, K 125, 144 [41] Fukuhara, M 219, 220, 236 [47] Furter, J.E 384, 459 [13] Da Prato, G 70, 74 [5] Dafermos, C.M 327, 350 [35] Dancer, E.N 316, 318, 320–322, 332, 350 [36]; 350 [37]; 350 [38]; 392, 397, 458, 459 [8]; 459 [9]; 459 [10] Dancer, N 107, 109, 143 [27] de Leenheer, P 302, 350 [41] de Mottoni, P 314, 329–331, 350 [42]; 438, 454, 460 [29] de Pagter, B 349 [27] DeAngelis, D.L 242, 350 [39] Degiovanni, M 121, 143 [28] Deimling, K 350 [40] Delgado, M 438, 459 [11] Ding, Y.H 97, 129, 132, 135, 136, 138, 140, 143 [12]; 143 [13]; 143 [29]; 143 [30]; 143 [31] Dlotko, T 332, 345, 349 [25] Dugundji, J 482, 483, 554 [14] Dunford, N 201, 234, 236 [40] Gamkrelidze, R.V 16, 29, 74 [1]; 75 [23]; 75 [30]; 223, 237 [83] Gao, Y 547, 554 [20] Garcia-Huidobro, M 554, 554 [17] Gatzke, H 297, 348 [1] Gautier, S 228, 236 [48] Ge, W 540–542, 546, 547, 553, 554, 554 [19]; 554 [20]; 555 [23]; 555 [41]; 555 [42]; 556 [65]; 556 [66]; 557 [85] Gelfand, I.M 4, 8, 75 [24] Ginzburg, V.L 110, 113, 144 [43]; 144 [44]; 144 [45] Girardi, M 115, 136, 138, 143 [30]; 144 [46] Girsanov, I.V 161, 236 [49] Glas, D 254, 350 [55] Gonzales, R 223, 236 [50] Goursat, E 223, 236 [51] Graef, J 542, 554 [18] Granas, A 482, 483, 554 [14] Greiner, G 350 [52] Grossberg, S 297, 350 [53]; 350 [54] Guckenheimer, J 80, 138, 142, 144 [47] Guo, D 519, 525, 536, 554 [21] Guo, Y.X 121, 144 [48]; 541, 542, 546, 547, 554 [19]; 554 [20]; 554 [22]; 555 [23]; 556 [64] Gupta, C.P 481, 489, 494, 495, 497, 505, 513, 516, 542, 554, 554 [1]; 554 [13]; 554 [17]; 555 [24]; 555 [25]; 555 [26]; 555 [27]; 555 [28]; 555 [29]; 555 [30]; 555 [31]; 555 [32]; 555 [33]; 555 [34]; 555 [35]; 555 [36]; 555 [37]; 555 [38]; 555 [39]; 555 [40] Eden, A 350 [43] Eilbeck, J.C 363, 384, 439, 452, 459 [12]; 459 [13] Ekeland, I 15, 50, 74 [13]; 75 [21]; 80, 115, 118, 126, 135, 139, 140, 143 [25]; 143 [32]; 143 [33]; 143 [34]; 143 [35] Elsner, L 286, 350 [44] Evans, L.C 68, 70, 74 [17] Fadell, E.R 88, 109, 121, 143 [36] Author Index Gürel, B.Z 113, 144 [44] Haddad, A 297, 348 [1] Haddad, G 227, 236 [52] Hadeler, K 254, 350 [55] Hájek, O 223, 236 [53] Hale, J.K 301–303, 305, 307, 309, 313, 314, 329, 331, 332, 344, 349 [24]; 349 [26]; 350 [56]; 350 [57]; 350 [58]; 350 [59]; 350 [60]; 350 [61]; 439, 459 [14] Hall, G.R 80, 145 [75] Han, Y 482, 557 [93] Hartman, P 151, 152, 220, 236 [54]; 236 [55] He, X 540, 553, 555 [41]; 555 [42] Heijmans, H 349 [27] Henderson, J 553, 554 [3] Henry, D 332–334, 350 [62] Herman, M 113, 144 [49] Hess, P 314, 316, 318, 320, 321, 329, 332, 348 [3]; 350 [38]; 350 [63]; 350 [64]; 350 [65]; 439, 459 [15] Hilbert, D 7, 8, 74 [16]; 223, 236 [37]; 314, 349 [34] Hirsch, M.W 242, 247, 249, 250, 254–256, 258, 264, 265, 269, 271, 272, 275, 281, 291, 292, 296, 297, 299, 300, 302, 317, 319, 349 [16]; 349 [17]; 351 [66]; 351 [67]; 351 [68]; 351 [69]; 351 [70]; 351 [71]; 351 [72]; 351 [73]; 351 [74]; 351 [75]; 351 [76]; 351 [77]; 351 [78]; 351 [79]; 351 [80] Hobson, E.W 175, 236 [56] Hofbauer, J 301, 302, 351 [81]; 351 [82] Hofer, H 80, 113, 115, 125, 144 [50]; 144 [51]; 144 [52] Holmes, P 80, 138, 142, 144 [47] Hopf, H 315, 348 [2] Howard, P 156, 236 [57] Hsu, S.-B 302, 320, 332, 351 [83]; 351 [84] Il’in, V 470, 555 [43]; 555 [44] Infante, G 532, 542, 555 [45]; 555 [46]; 555 [47] Ishii, H 70, 75 [18]; 75 [19]; 223, 236 [58] Isidori, A 3, 75 [25] Izydorek, M 121, 124, 144 [53]; 144 [54] Jeanjean, L 133, 144 [55] Jentzsch, R 315, 351 [85] Jiang, J 265, 296, 314, 319, 332, 351 [86]; 351 [87]; 351 [88]; 351 [89]; 351 [90]; 352 [120]; 352 [121]; 356 [229]; 356 [230]; 356 [231] Jurdjevic, V 3, 67, 75 [26] 561 Kamke, E 223, 237 [59]; 241, 285, 351 [91] Karakostas, G.L 542, 547, 556 [48]; 556 [49]; 556 [50]; 556 [51]; 556 [52]; 556 [53] Kato, T 274, 275, 351 [92] Keller, H.B 314, 351 [93]; 352 [94]; 352 [95] Kenmochi, N 181, 237 [60] Kerman, E 110, 144 [45] Kerscher, W 352 [96] Kneser, H 219, 221, 237 [61] Kobayashi, J 213, 237 [62] Kolmogorov, A 297, 352 [97] Krasnosel’skii, M.A 88, 144 [56]; 314, 323, 324, 352 [98]; 352 [99]; 352 [100]; 352 [101]; 363, 411, 459 [16]; 519, 556 [54] Krause, U 315, 323, 324, 327, 352 [102]; 352 [103] Krein, M.G 274, 315, 352 [104] Krisztin, T 303, 313, 352 [105]; 352 [106]; 352 [107]; 352 [108] Kryszewski, W 95, 106, 121, 136, 144 [57]; 144 [58] Kucia, A 190, 237 [63] Kunisch, K 305, 352 [109] Kunze, H 289, 352 [110]; 352 [111]; 352 [112] Kuznetsov, Y.A 328, 352 [113] Ladyzhenskaya, L 314, 352 [100]; 352 [114] Lakshmikantham, V 465, 489, 519, 525, 536, 554 [12]; 554 [21]; 555 [31] LaSalle, J.P 316, 352 [116] Lasry, J.M 115, 121, 143 [20]; 143 [35]; 190, 235 [10] Lassoued, L 115, 143 [34] Lazer, A.C 332, 350 [64]; 439, 459 [2]; 459 [3]; 459 [15] Ledyaev, Yu.S 175, 226, 228, 236 [33]; 236 [34]; 237 [64] Legget, R 536, 548, 556 [55] Lemmens, B 315, 352 [117] Lemmert, R 284, 352 [118] Leung, A.W 352 [119] Levinson, N 16, 74 [15]; 349 [28] Li, G 96, 144 [59] Li, M 301, 352 [115] Li, S.J 83, 86, 91, 106, 118, 121, 144 [60]; 144 [61]; 144 [62] Li, W 301, 357 [237] Li, Y 54, 75 [36] Li, Y.Q 92, 144 [63] Liang, X 314, 332, 352 [120]; 352 [121] Lieb, E.H 129, 144 [64] Lions, J.-L 3, 75 [27] Lions, P.-L 68, 70, 74 [17]; 75 [18]; 75 [19]; 75 [20]; 95, 129, 144 [65]; 233, 236 [39] 562 Author Index Liu, B 516, 517, 525, 529, 532, 541, 556 [56]; 556 [57]; 556 [58]; 556 [59]; 556 [60]; 556 [61]; 556 [62] Liu, C.G 115, 144 [67] Liu, J.Q 83, 91, 106, 121, 124, 144 [60]; 144 [61]; 144 [66] Liu, X 541, 542, 547, 554 [22]; 556 [63]; 556 [64] Liu, Y 542, 553, 556 [65]; 556 [66] Loewy, R 353 [122] Long, Y 80, 106, 115, 118, 121, 144 [67]; 145 [68]; 145 [69]; 145 [70]; 145 [71] López-Gómez, J 363, 384, 390, 392, 394–397, 404, 411, 438, 439, 452, 458, 459 [10]; 459 [11]; 459 [12]; 459 [13]; 459 [17]; 459 [18]; 459 [19]; 459 [20]; 459 [21]; 459 [22]; 459 [23]; 459 [24]; 459 [25]; 459 [26] Loss, M 129, 144 [64] Lui, R 315, 353 [123] Lunardi, A 332, 334, 339–342, 345, 353 [124] Lupo, D 92, 94, 124, 144 [42] Lupulescu, V 154, 237 [65] Lyapunov, A.M 106, 145 [72] Ma, R 475, 525, 532, 533, 535, 540–542, 556 [67]; 556 [68]; 556 [69]; 556 [70]; 556 [71]; 556 [72]; 556 [73]; 556 [74]; 557 [75]; 557 [76]; 557 [77]; 557 [78]; 557 [79] Manasevich, R 554, 554 [17] Mancini, G 115, 121, 143 [20] Marano, S 481, 489, 557 [80]; 557 [81] Marsden, J.E 6, 19, 75 [28] Martin, R.H 305, 332, 335–338, 343, 353 [125]; 353 [126]; 353 [127]; 353 [128]; 353 [129]; 353 [130] Martin, R.H., Jr 153, 154, 237 [66] Matano, H 247, 314, 320, 321, 332, 348 [3]; 349 [23]; 353 [131]; 353 [132]; 353 [133]; 353 [134]; 353 [135] Matzeu, M 115, 144 [46] Mawhin, J 80, 89, 124, 144 [40]; 145 [73]; 363, 459 [27]; 460 [28]; 474, 476, 477, 486, 490, 491, 498, 557 [82] May, R 315, 353 [136] McDuff, D 80, 145 [74] McShane, E.J 293, 353 [137] Meyer, K.R 80, 145 [75] Michael, E 212, 229, 237 [67]; 237 [68] Mierczy´nski, J 265, 280, 296, 327, 328, 353 [138]; 353 [139]; 353 [140]; 353 [141]; 353 [142]; 353 [143] Miller, R.K 320, 357 [236] Mincheva, M 345, 353 [144]; 353 [145] Mischaikow, K 353 [146] Mishchenko, E.F 29, 75 [30]; 223, 237 [83] Moiseev, E 470, 555 [43]; 555 [44] Molina-Meyer, M 363, 390, 459 [21]; 459 [22]; 459 [23]; 459 [24] Monteiro Marques, M.D.P 190, 227, 228, 231, 235 [21]; 235 [22]; 235 [23] Mora, X 332, 340, 353 [147] Mordukhovich, B.S 153, 237 [69] Moreau, J 13, 75 [29] Moser, J 106, 107, 145 [76] Moshinsky, M 470, 557 [83] Muldowney, J 301, 352 [115] Müller, M 241, 285, 353 [148] Murty, K 475, 557 [84] Nagel, R 352 [96] Nagumo, M 151, 166, 237 [70] Necula, M 154, 163, 178, 231, 233, 235 [24]; 237 [65]; 237 [71]; 237 [72]; 237 [73]; 237 [74] Neumann, M 282, 284, 286–289, 349 [18] Ni, X 554, 557 [85] Nicolaenko, B 350 [43] Ntouyas, S.K 469, 481, 494, 497, 505, 513, 555 [32]; 555 [33]; 555 [34]; 555 [35]; 555 [36]; 557 [86] Nussbaum, R.D 315, 327, 352 [102]; 352 [117]; 354 [149]; 354 [150]; 354 [151] Ohta, Y 305, 354 [152] Olian Fannio, L 121, 143 [28] Ono, K 125, 144 [41] Ortega, R 302, 354 [153]; 439, 447, 458, 459 [5]; 459 [10]; 459 [25]; 460 [30] Oster, G 315, 353 [136] Oxtoby, J 257, 354 [154] Palamides, P.K 547, 557 [87] Palmer, K.J 138, 140, 145 [77] Pao, C.V 341, 343, 354 [155] Pardo, R 458, 459 [26] Pavel, N.H 230–232, 234, 235 [5]; 237 [75]; 237 [76]; 237 [77]; 237 [78]; 237 [79] Pazy, A 333, 354 [156] Peano, G 149, 237 [80] Peixoto, M 246, 354 [157] Perron, O 223, 237 [81]; 315, 354 [158] Pituk, M 313, 354 [159] Plaskacz, S 190, 227, 228, 236 [45]; 237 [82] Plemmons, R 287–289, 349 [19] Poláˇcik, P 280, 281, 314, 320, 329, 332, 343, 345, 349 [22]; 350 [65]; 354 [160]; 354 [161]; 354 [162]; 354 [163]; 354 [164]; 354 [165] Pontryagin, L.S 29, 75 [30]; 223, 237 [83] Post, W.M 242, 350 [39] Author Index Precupanu, Th 11, 47, 74 [6] Protter, M.H 346, 354 [166] Przeradski, B 557 [88] Qian, C 542, 554 [18] Qiu, J 541, 542, 547, 554 [22]; 556 [64] Quincampoix, M 161, 237 [84] Rabinowitz, P.H 79, 80, 82, 87, 88, 99, 100, 109, 111, 118, 121, 136, 138, 139, 143 [19]; 143 [21]; 143 [26]; 143 [36]; 145 [78]; 145 [79]; 145 [80]; 145 [81]; 363, 371, 392, 395–397, 404, 407, 459 [6]; 460 [31] Raffoul, Y 535, 557 [89] Ramos, M 92, 94, 124, 144 [42] Ranft, P 323, 324, 327, 352 [103] Ratiu, T.S 6, 19, 75 [28] Raugel, G 354 [167] Redheffer, R 153, 154, 237 [85] Rescigno, A 297, 354 [168] Richardson, I 297, 354 [168] Rishel, R.W 3, 73, 75 [22] Robinson, C 349 [13]; 354 [169] Robinson, J 354 [170] Rockafellar, R.T 11, 14, 47, 75 [31]; 75 [32] Roxin, E 174, 237 [86] Rubin, J.E 156, 236 [57] Rudin, W 306, 354 [171] Ruelle, D 354 [172] Ruf, B 115, 121, 143 [20] Rutman, M.A 274, 315, 352 [104]; 354 [173] Rybicki, S 107, 109, 143 [27] Rze˙zuchowski, T 190, 228, 236 [45] Sachkov, Y.L 3, 16, 18, 57, 67, 74 [2] Sacker, R 332, 354 [174] Salamon, D 80, 145 [74] Samarskii, A 470, 554 [8] Sánchez, L 302, 354 [153] Sandholm, W 302, 351 [81] Sandstede, B 314, 349 [22] Saperstone, S.H 246, 354 [175] Sattinger, D 314, 354 [176] Schappacher, W 305, 352 [109] Schiaffino, A 314, 329–331, 350 [42]; 438, 454, 460 [29] Schneider, H 282, 284, 286, 301, 353 [122]; 354 [177] Schuur, I.D 227, 235 [6] Schwartz, J.T 201, 234, 236 [40] Scorza Dragoni, G 190, 237 [87]; 237 [88] Secchi, S 135, 145 [82] 563 Selgrade, J 297, 315, 354 [178]; 355 [179]; 355 [180]; 355 [181] Sell, G 332, 354 [174] Séré, E 126, 135, 139, 140, 142, 143 [25]; 145 [83] Severi, F 159, 222, 237 [89]; 238 [90] Shan, W 541, 555 [23] Shi, S.Z 231, 232, 238 [91] Siegel, D 289, 345, 352 [110]; 352 [111]; 352 [112]; 353 [145] Sivasundaram, S 475, 557 [84] Sleeman, B.D 332, 357 [246] Slemrod, M 327, 350 [35] Smale, S 294, 297, 351 [80]; 355 [182] Smith, H.L 242, 247, 249, 250, 255, 266, 267, 269, 271, 272, 281, 293, 296, 297, 299, 301, 302, 304, 305, 311–315, 318, 320, 327–329, 332, 337, 343, 345, 346, 350 [41]; 351 [83]; 353 [129]; 353 [130]; 353 [146]; 355 [183]; 355 [184]; 355 [185]; 355 [186]; 355 [187]; 355 [188]; 355 [189]; 355 [190]; 355 [191]; 355 [192]; 355 [193]; 355 [194]; 355 [195]; 355 [196]; 355 [197]; 355 [198]; 355 [199]; 355 [200]; 355 [201]; 355 [202]; 355 [203]; 357 [248]; 439, 460 [32]; 460 [33] Smith, R.A 302, 355 [204] Smoller, J 345, 355 [205] So, J.W.-H 301, 351 [82] Somolinos, A.S 314, 329, 332, 350 [60]; 439, 459 [14] Sontag, E.D 3, 75 [33] Spanier, E.H 83, 145 [84] Staicu, V 233, 235 [17] Stanczy, R 557 [88] Stern, R.J 175, 226, 236 [33]; 236 [34]; 282, 284–289, 349 [18]; 355 [206] Stewart, H 355 [207] Struwe, M 87, 113, 145 [85]; 145 [86] Stuart, C.A 126, 131, 135, 145 [82]; 145 [87] Suárez, A 438, 459 [11] Szlenk, W 142, 145 [88] Szulkin, A 86, 90, 92, 95, 96, 106, 109, 115, 118, 121, 124, 129, 135, 136, 138, 142 [6]; 144 [57]; 144 [58]; 144 [59]; 144 [62]; 145 [89]; 145 [90]; 145 [91]; 145 [92]; 145 [93]; 145 [94] Takáˇc, P 274, 281, 315, 316, 318, 321, 322, 327, 328, 332, 350 [45]; 350 [46]; 356 [208]; 356 [209]; 356 [210]; 356 [211]; 356 [212]; 356 [213]; 356 [214]; 356 [215]; 356 [216] Takahashi, T 181, 237 [60] Tallos, P 228, 238 [92] Tanaka, K 135, 145 [95] 564 Author Index Tang, M 350 [50] Temam, R 350 [43] Terešˇcák, I 329, 354 [164]; 354 [165]; 356 [217] Thieme, H.R 247, 250, 255, 266, 267, 269, 271, 272, 281, 304, 313, 315, 320, 332, 353 [146]; 355 [197]; 355 [198]; 355 [199]; 355 [200]; 355 [201]; 356 [218] Thompson, A.C 324, 356 [219] Tikhomirov, V.M 4, 75 [34] Timoshenko, S 470, 557 [90] Tineo, A 296, 356 [220]; 439, 447, 458, 459 [25]; 460 [30]; 460 [34]; 460 [35] Travis, C.C 242, 350 [39] Treiman, J.S 166, 238 [93] Trofimchuk, S 494, 495, 555 [37]; 555 [38]; 555 [39]; 555 [40] Tsamatos, P.Ch 469, 481, 494, 497, 505, 513, 542, 547, 555 [32]; 555 [33]; 555 [34]; 555 [35]; 555 [36]; 556 [48]; 556 [49]; 556 [50]; 556 [51]; 556 [52]; 556 [53]; 557 [86] Wang, Z.-Q 125, 143 [14]; 143 [15]; 144 [39] Wax, N 297, 348 [1] Wa˙zewski, T 227, 238 [107] Webb, J.R.L 497, 505, 532, 542, 554 [15]; 554 [16]; 555 [46]; 555 [47]; 557 [92] Weinberger, H.F 315, 346, 354 [166]; 356 [232] Weinstein, A 106, 107, 111, 146 [98]; 146 [99] Wiggins, S 138, 140, 146 [100] Wilder, R 322, 356 [233] Willem, M 80–82, 89, 90, 92, 94, 99, 124, 127, 129, 132, 135, 140, 143 [16]; 143 [31]; 144 [42]; 145 [73]; 146 [101]; 146 [102] Williams, L 536, 548, 556 [55] Wolenski, P.R 226, 236 [34] Wolkowicz, H 285, 355 [206] Wu, J 303, 313, 320, 352 [105]; 352 [106]; 352 [107]; 352 [108]; 356 [234]; 356 [235]; 357 [236] Wysocki, K 115, 144 [50] Uhl, R 282, 356 [221] Ursescu, C 151, 152, 154, 158, 161, 162, 166, 175, 176, 210, 213, 222, 225, 227, 228, 234, 235 [25]; 235 [26]; 237 [77]; 238 [94]; 238 [95]; 238 [96]; 238 [97]; 238 [98]; 238 [99]; 238 [100]; 238 [101] Uryson, P.S 314, 356 [222] Xiao, D 301, 357 [237] Xu, J.-M 489, 554 [1] Xu, X 118, 135, 138, 145 [70]; 146 [103] van den Driessche, P 296, 301, 356 [223] van Duijn, C 349 [27] Verduyn Lunel, S.M 302, 303, 305, 307, 309, 315, 350 [61]; 352 [117] Vespri, V 340, 349 [29] Vidyasagar, M 282, 284, 286, 301, 354 [177] Vinter, R.B 73, 74 [14]; 223, 236 [35] Viterbo, C 113, 115, 145 [96]; 146 [97] Volkmann, P 282, 284, 352 [118]; 356 [224] Vrabie, I.I 21, 75 [35]; 151, 152, 154, 171, 173, 174, 177, 178, 228, 230–234, 235 [24]; 235 [27]; 235 [28]; 237 [74]; 237 [78]; 237 [79]; 238 [102]; 238 [103]; 238 [104]; 238 [105]; 238 [106] Vulikh, B 338, 356 [225] Walcher, S 282, 356 [226] Walter, W 282, 289, 343, 356 [227]; 356 [228] Walther, H.-O 303, 313, 352 [105] Waltman, P 249, 302, 320, 332, 343, 351 [83]; 351 [84]; 355 [202]; 355 [203] Wang, H.Z 54, 75 [36]; 532, 557 [79] Wang, J 533, 553, 557 [91]; 557 [94]; 557 [95] Wang, Y 296, 314, 332, 356 [229]; 356 [230]; 356 [231] Yang, B 542, 554 [18] Yorke, J.A 151–154, 217, 220, 238 [108]; 238 [109] Yoshizawa, T 216, 238 [110] Yosida, S 47, 75 [37] Yu, J 516, 556 [61]; 556 [62] Yu, S 265, 319, 351 [90] Zabczyk, J 3, 67, 75 [38] Zabreiko, P.P 314, 352 [101] Zanolin, F 332, 357 [238]; 439, 460 [36] Zaremba, S.C 227, 238 [111] Zeeman, E.C 296, 301, 357 [239]; 357 [240]; 357 [241]; 357 [242] Zeeman, M.L 296, 301, 356 [223]; 357 [240]; 357 [241]; 357 [242]; 357 [243] Zehnder, E 80, 104, 106, 113, 115, 121, 124, 125, 142 [3]; 142 [4]; 143 [24]; 144 [50]; 144 [51]; 144 [52] Zeidler, E 274, 357 [244] Zhang, M 482, 557 [93] Zhang, Z 533, 557 [94]; 557 [95] Zhao, X.-Q 314, 332, 357 [245]; 357 [246] Zheng, D 553, 557 [91] Zhu, C 115, 144 [67]; 145 [71] Zhu, H.-R 302, 357 [247]; 357 [248] Ziehe, M 315, 355 [181] Zou, W 121, 135, 145 [93]; 145 [94] Subject Index τ -topology, 95 component of coexistence states, 404 component of positive solutions, 394 concave functional, 536 conditionally completely continuous map, 266 cone, 536 – conjugate, 163 – contingent, 160 – in the sense of Bony, 162 – normal, 11, 15 – proximal normal, 161 – tangent, 15 conjugate mapping, 141 conjugate point, 8, 67 Conley–Zehnder index, 106 contingent derivative, 227 contraction, 325 control, – bang-bang, 9, 47 – closed loop, 62 – constraints, – feedback, 62 – open loop, 62 – optimal, 9, 55, 62 convergent, 246, 317 cooperative map, 329 cost functional, critical orbit, 88 crossing number, 393 curve of change of stability, 376–378 curve of neutral stability, 376, 377 accessible point, 244 adapted space, 335 admissible pair, Arnold conjecture, 125 asymptotically order stable, 262 attainable set, 55 attract, 245 attractor, 245 axiom – of choice, 156 – of countable choice, 156 – of dependent choice, 156 Banach lattice, 338 basin, 258 Bernoulli shift, 140 Bochner integral, 21 Bolza problem, 9, 30, 40, 56 Brezis–Browder ordering principle, 157 C0 -semigroup, 229 Carathéodory function, 471 Cartan formula, 24 Cerami sequence, 96 chain recurrent, 293 chronological exponential – left, 22 – right, 19 classical solution, 468 closed loop system, 62 closed order interval, 244 cocycle identities, 333 coexistence state, 361, 401 compact map, 335 comparison property – Carathéodory, 198 – of a function, 180 – of a multifunction, 212 competing species, 379, 438 competitive map, 329 completely continuous map, 266 deformation lemma, 81 derivation, 18 derivative – contingent, 227 – Dini right lower, 175 – right directional – – of the norm, 162 diameter, 258 directional derivative, 11, 14 domain, 149 doubly accessible point, 257 dual cone, 281 565 566 eigenvalue, 393 Ekeland variational principle, 15 entire orbit, 320 equation – Bellman, 63 – dynamic programming, 63 – eikonal, 72 – Euler–Lagrange, – generalized Bellman, 224 – Hamilton, 6, 27 – Hamilton–Jacobi, 7, 63 – semilinear differential, 230 equilibrium, 245, 316 equivariant mapping, 87 escape time, 334 eventually strongly monotone, 247 eventually strongly monotone map, 316 extremal, falling interval, 249, 317 five functional fixed point theorem, 548 fixed point, 316 – index, 393 – index theorem, 525 – set, 87 Floquet multipliers, 375, 377, 415 flow, 17, 245 – Hamiltonian, 27 function – Carathéodory comparison, 197 – comparison, 154 – conjugate, 12 – convex, 10 – effective domain, 10 – epigraph, 10 – integrable, 21 – locally Lipschitz, 14 – lower semicontinuous, 10 – of Carathéodory type, 190 – positively sublinear, 173 – proper, 10 – regulated, 229 – Severi differentiable, 222 – strongly measurable, 21 – synthesis, 63 – weakly measurable, 21 functional ordering, 245 fundamental matrix, 414 generalized gradient, 14 genus, 86 geometrically distinct homoclinic solutions, 128 geometrically distinct periodic solutions, 100 global attractor, 246 Subject Index global process, 333 group representation, 87 Hamiltonian, 6, 27 Hamiltonian lift, 28 high intensity competition, 384 homoclinic solutions, 126 – geometrically distinct, 128 hyperbolic point solution, 126 hysteresis point, 431 index, 88 infimum, 244 infinitesimal generator, 229 invariant, 245 – forward, 152 – local, 152 – locally right, 152 – mapping, 87 – right, 152 – set, 87 invariantly connected set, 316 isometric group representation, 87 k-bump solution, 138 K-competitive, 289 K-cooperative, 285 K-irreducible matrix, 288 K-positive matrix, 284 Krasnoselskii’s fixed point theorem, 519 Krein–Rutman operator, 274 Lagrange problem, 9, 56 Lebesgue point, 34 Legendre transform, 6, 12 Legget–Williams fixed point theorem, 536 lemma of Cârj˘a–Ursescu, 177, 210 limit genus, 92 limit index, 92 Liouville formula, 414 Lipschitz system, 292 local linking, 83 local semiflow, 251 locally closed, 335 locally closed set, 86 locally Hölder map, 333 locally Lipschitz map, 333 locally monotone map, 290 locally strongly monotone map, 290 logistic equation, 364 Lotka–Volterra periodic system, 362 low intensity competition, 384 lower boundary, 244 Subject Index M-maximal element, 157 m-point BVP, 470 mapping – positively sublinear, 208 Maslov index, 106 maximal element, 244 Mayer problem, 9, 56 mild solution, 334, 468 minimal element, 244 minimal set, 269 monotone map, 316 monotone system, 282 multi-point BVP, 470 multibump solution, 138 multifunction, 164 – l.s.c at a point, 200 – l.s.c on a set, 200 – u.s.c at a point, 200 – u.s.c on a set, 200 necessary conditions, 5, 29 – Jacobi, – Legendre, – Weierstrass–Erdmann, negative semiorbit, 289 non-cooperative periodic system, 456 nonexpansive map, 325 nonlocal BVP, 470 nonlocal condition, 463 nonlocal initial value problem, 465 nonresonance, 489 nonwandering set, 319 normality constant, 259 normally ordered space, 259 omega limit set, 246 open order interval, 244 operator pencil, 393 optimal control problem, optimal pair, optimal state, orbit, 87, 245, 333 – of a set, 266 order bounded, 244 order compact, 322 order neighborhood, 261 order norm, 261 order related, 244 order stable, 262 order topology, 261 ordered Banach space, 245 ordered space, 243 ordered subspace, 244 p-convex, 245 Palais–Smale condition, 81 Palais–Smale sequence, 81 period, 245 periodic, 245, 316 periodic solutions – geometrically distinct, 100 Poincaré map, 377, 414, 417, 441 point dissipative, 329 Poisson bracket, 27 polyhedral, 281 positive operator, 316 positive semiorbit, 289 positive solution, 519 positively invariant, 245, 333 positivity properties, 364 predator–prey model, 386, 455 preorder, 176 – closed, 176 principal characteristic exponent, 428, 449 principal eigenvalue, 417 problem – abnormal, 30 – normal, 30 – of local invariance, 152 process, 333 projection – subordinated to V, 158 proposition – Roxin, 174 proximal neighborhood, 158 (PS)-attractor, 96 (PS)c -condition, 81 (PS)∗ -condition, 91 (PS)∗ -sequence, 91 pseudo-gradient vector field, 82 pseudo-symmetric function, 547 quasi-cooperative system, 440 quasiconvergent, 246, 317 quasiderivative map, 273 quasimonotone condition, 282 quasimonotone condition (QMD), 304 reachable set, 55 reaction–diffusion systems, 339 recurrence, 293 regularization, 13 relation – preorder, 156 relative Morse index, 105, 106 representation – diffeomorphisms, 18 567 568 – points, 18 – tangent vectors, 18 – vector fields, 18 reproducing cone, 322 resolvent operator, 366 resonance, 489 rising interval, 249, 317 sectorial, 333 semi-trivial positive state, 361 semiconjugate mapping, 141 semiflow, 245 semiorbit, 251 set – reachable, 224 – target, 224 Severi differential – at x, 222 – at x in the direction u, 222 singular value of an operator pencil, 393 solution – ε-approximate, 168, 206 – Carathéodory, 190 – contingent, 227, 233 – flow, 245 – global, 172 – mild, or C , 229 – noncontinuable, 172, 207 – of a differential inclusion, 202 – of inequality, 224 – process, 335 – right funnel, 219 – semigroup, 229 – to a differential equation, 149 spectral radius, 274 stable fixed point, 320 stable from above, 258 stable from below, 258 standard cone, 245 state of the system, strictly monotone, 246 strictly monotone map, 316 strong approximation, 265 strong maximum principle, 393 strong ordering, 244 strongly accessible, 265 strongly increasing, 370 strongly monotone, 246 strongly negative, 366 strongly order-preserving, 246 strongly order-preserving (SOP) map, 316 strongly positive, 366 strongly positive operator, 316 strongly sublinear map, 324 Subject Index sub-critical turning point, 431 subdifferential, 11 subgradient, 11 sublinear map, 323 subset – Carathéodory invariant, 197 – Carathéodory locally invariant, 197 – closed relative to D, 151 – flow invariant, 153 – invariant, 178, 210 – locally closed, 151, 158 – locally invariant, 178, 210 – right viable with respect to f , 149 – right viable with respect to F , 202 – viable, 166 subsolution, 367, 368 super-critical turning point, 431 superlinear indefinite model, 388 supersolution, 367, 368 supremum, 244 symbiotic species, 387, 411 synthesis problem, 63 system – strongly decreasing, 226 – weakly decreasing, 226 t -cross-section of Fτ,ξ , 219 theorem – Bony, 153 – Brezis–Browder, 157 – Hukuhara, 220 – Kneser, 221 – Nagumo, 151, 166 – Pavel, 231 – Peano, 149 – Redheffer, 154 – Scorza Dragoni, 190 – Shi, 232 – Vitali, 234 – Yorke, 217, 220 – Yoshizawa, 216 topological degree, 393 topologically equivalent, 292 totally competitive, 294 totally ordered arc, 255 trajectory, 245, 333 transversality conditions, 7, 30, 61 trivial state, 361 uniform global attractor, 321 unilateral bifurcation theorem, 397 unordered, 244 unstable from above, 268 Subject Index unstable from below, 269 upper boundary, 244 variation of constants formula, 24 variation of parameters method, 24 vector – metric normal to K at ξ , 153, 161 – tangent in the sense of – – Bony, 162 – – Bouligand–Severi, 159 – – Clarke, 163 569 – – Federer, 161 vector field – complete, 17 – Hamiltonian, 27 – nonautonomous, 16 vector ordering, 245 very strongly order preserving (VSOP) process, 336 viability, 149 Yosida approximation, 13 This page intentionally left blank ... 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... elegant articles This volume consists of six chapters covering a variety of problems in ordinary differential equations Both, pure mathematical research and real word applications are reflected pretty... relation with the practical applications of the theory Its evolution has shown that its methods and tools are drawn from a large spectrum of mathematical branches such as ordinary differential equations,