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beam equation; system of beam wave equation; initial boundary value problem; bifurcation; Fučík spectrum
In this paper we present several nonlinear models of suspension bridges; most of them have been introduced by Lazer and McKenna. We discuss some results which were obtained by the authors and other mathematicians for the boundary value problems and initial boundary value problems. Our intention is to point out the character of these results and to show which mathematical methods were used to prove them instead of giving precise proofs and statements.
[1] N. U. Ahmed, H. Harbi: Mathematical analysis of dynamic models of suspension bridges. SIAM J. Appl. Math. 58 (1998), 853–874. DOI 10.1137/S0036139996308698 | MR 1616611
[2] J. M. Alonso, R. Ortega: Global asymptotic stability of a forced Newtonian system with dissipation. Journal of Math. Anal. and Appl. 196 (1995), 965–986. DOI 10.1006/jmaa.1995.1454 | MR 1365234
[3] J. Berkovits, P. Drábek, H. Leinfelder, V.  Mustonen and G. Tajčová: Time-periodic oscillations in suspension bridges: existence of unique solution. Nonlin. Analysis: Real Word Appl. 1 (2000), 345–362. MR 1791531
[4] J. Berkovits, V.  Mustonen: Existence and multiplicity results for semilinear beam equations. Colloquia Mathematica Societatis János Bolyai Budapest, 1991, pp. 49–63. MR 1468743
[5] J. Čepička: Numerical experiments in nonlinear problems. PhD. Thesis, University of West Bohemia, Pilsen, 2002. (Czech)
[6] Y. Chen, P. J. McKenna: Travelling waves in a nonlinear suspended beam: theoretical results and numerical observations. Journal of Diff. Eq. 136 (1997), 325–355. DOI 10.1006/jdeq.1996.3155 | MR 1448828
[7] Q. H. Choi, K.  Choi, T.  Jung: The existence of solutions of a nonlinear suspension bridge equation. Bull. Korean Math. Soc. 33 (1996), 503–512. MR 1424092
[8] Q. H. Choi, T.  Jung, P. J.  McKenna: The study of a nonlinear suspension bridge equation by a variational reduction method. Applicable Analysis 50 (1993), 73–92. DOI 10.1080/00036819308840185 | MR 1281204
[9] Y. S. Choi, K. S.  Jen, P. J.  McKenna: The structure of the solution set for periodic oscillations in a suspension bridge model. IMA Journal of Applied Math. 47 (1991), 283–306. DOI 10.1093/imamat/47.3.283 | MR 1141492
[10] P. Drábek: Jumping nonlinearities and mathematical models of suspension bridges. Acta Math. Inf. Univ. Ostraviensis 2 (1994), 9–18. MR 1309060
[11] P. Drábek, G. Holubová: Bifurcation of periodic solutions in symmetric models of suspension bridges. Topological Methods in Nonlin. Anal. 14 (1999), 39–58. DOI 10.12775/TMNA.1999.021
[12] P. Drábek, H.  Leinfelder, G.  Tajčová: Coupled string-beam equations as a model of suspension bridges. Appl. Math. 44 (1999), 97–142. DOI 10.1023/A:1022257304738 | MR 1667633
[13] P. Drábek, P.  Nečesal: Nonlinear scalar model of suspension bridge: existence of multiple periodic solutions. Nonlinearity 16 (2003), 1165–1183. DOI 10.1088/0951-7715/16/3/320 | MR 1975801
[14] J. Dupré: Bridges. Black Dog & Levenathal Publishers, New York, 1997.
[15] A. Fonda, Z.  Schneider, F.  Zanolin: Periodic oscillations for a nonlinear suspension bridge model. J.  Comput. Appl. Math. 52 (1994), 113–140. DOI 10.1016/0377-0427(94)90352-2 | MR 1310126
[16] J. Glover, A. C.  Lazer, P. J. McKenna: Existence and stability of large scale nonlinear oscillations in suspension bridges. J. Appl. Math. Physics (ZAMP) 40 (1989), 172–200. DOI 10.1007/BF00944997 | MR 0990626
[17] G. Holubová, A. Matas: Initial-boundary value problem for nonlinear string-beam system. J. Math. Anal. Appl, Accepted. MR 2020197
[18] L. D. Humphreys: Numerical mountain pass solutions of a suspension bridge equation. Nonlinear Analysis 28 (1997), 1811–1826. DOI 10.1016/S0362-546X(96)00020-X | MR 1432634 | Zbl 0877.35126
[19] L. D. Humphreys, P. J.  McKenna: Multiple periodic solutions for a nonlinear suspension bridge equation. IMA Journal of Applied Math (to appear). MR 1739628
[20] D. Jacover, P. J. McKenna: Nonlinear torsional flexings in a periodically forced suspended beam. Journal of Computational and Applied Math. 52 (1994), 241–265. DOI 10.1016/0377-0427(94)90359-X | MR 1310133
[21] A. C. Lazer, P. J. McKenna: Fredholm theory for periodic solutions of some semilinear P.D.Es with homogeneous nonlinearities. Contemporary Math. 107 (1990), 109–122. DOI 10.1090/conm/107/1066474 | MR 1066474
[22] A. C. Lazer, P. J.  McKenna: A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. Proc. Amer. Math. Society 106 (1989), 119–125. DOI 10.1090/S0002-9939-1989-0942635-9 | MR 0942635
[23] A. C. Lazer, P. J. McKenna: Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. Trans. Amer. Math. Society 315 (1989), 721–739. DOI 10.1090/S0002-9947-1989-0979963-1 | MR 0979963
[24] A. C. Lazer, P. J.  McKenna: Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Review 32 (1990), 537–578. DOI 10.1137/1032120 | MR 1084570
[25] A. C. Lazer, P. J.  McKenna: Large scale oscillatory behaviour in loaded asymmetric systems. Ann. Inst. Henri Poincaré, Analyse non lineaire 4 (1987), 244–274. MR 0898049
[26] A. C. Lazer, P. J. McKenna: A symmetry theorem and applications to nonlinear partial differential equations. Journal of Diff. Equations 72 (1988), 95–106. DOI 10.1016/0022-0396(88)90150-7 | MR 0929199
[27] G. Liţcanu: A mathematical model of suspension bridges. Appl. Math (to appear). MR 2032147
[28] J. Malík: Oscillations in cable-stayed bridges: existence, uniqueness, homogenization of cable systems. J.  Math. Anal. Appl. 226 (2002), 100–126. MR 1876772
[29] J. Malík: Mathematical modelling of cable-stayed bridges: existence, uniqueness, continuous dependence on data, homogenization of cable systems. Appl. Math (to appear). MR 2032146
[30] J. Malík: Nonlinear oscillations in cable-stayed bridges. (to appear).
[31] A. Matas, J. Očenášek: Modelling of suspension bridges. Proceedings of Computational Mechanics  2, 2002, pp. 275–278.
[32] P. J. McKenna, K. S.  Moore: Mathematical arising from suspension bridge dynamics: Recent developements. Jahresber. Deutsch. Math.-Verein 101 (1999), 178–195. MR 1726743
[33] P. J. McKenna, W.  Walter: Nonlinear oscillations in a suspension bridge. Arch. Rational Mech. Anal. 98 (1987), 167–177. DOI 10.1007/BF00251232 | MR 0866720
[34] G. Tajčová: Mathematical models of suspension bridges. Appl. Math. 42 (1997), 451–480. DOI 10.1023/A:1022255113612 | MR 1475052
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