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nonlinearly coupled string-beam equation; periodic oscillations; jumping nonlinearities; degree theory
We consider nonlinearly coupled string-beam equations modelling time-periodic oscillations in suspension bridges. We prove the existence of a unique solution under suitable assumptions on certain parameters of the bridge.
[AO] J. M. Alonso, R. Ortega: Global asymptotic stability of a forced Newtonian system with dissipation. J. Math. Anal. Applications 196 (1995), 965–986. DOI 10.1006/jmaa.1995.1454 | MR 1365234
[BDL] J. Berkovits, P. Drábek, H. Leinfelder, V. Mustonen, G. Tajčová: Time-periodic oscillations in suspension bridges: existence of unique solutions. Nonlinear Analysis, Theory, Methods & Applications.
[CCJ] 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
[${\mathrm D}_1$] P. Drábek: Jumping nonlinearities and mathematical models of suspension bridges. Acta Math. Inf. Univ. Ostraviensis 2 (1994), 9–18. MR 1309060
[${\mathrm D}_2$] P. Drábek: Nonlinear noncoercive equations and applications. Z. Anal. Anwendungen 1 (1983), 53–65. MR 0720043
[FK] S. Fučík, A. Kufner: Nonlinear Differential Equations. Elsevier, Holland, 1980. MR 0558764
[FSZ] 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
[GLK] 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
[KJF] A. Kufner, O. John, S. Fučík: Function Spaces. Academia, Prague, 1977. MR 0482102
[${\mathrm LK}_1$] 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
[${\mathrm LK}_2$] 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
[${\mathrm LK}_3$] 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
[${\mathrm LK}_4$] 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
[${\mathrm LK}_5$] 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
[KW] 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
[PW] M. H. Protter, H. F. Weinberger: Maximum Principles in Differential Equations. Springer-Verlag New York, 1984. MR 0762825
[T] G. Tajčová: Mathematical models of suspension bridges. Appl. Math. 42 (1997), 451–480. DOI 10.1023/A:1022255113612 | MR 1475052
[V] O. Vejvoda et al.: Partial Differential Equations—time periodic solutions. Sijthoff Nordhoff, The Netherlands, 1981.
[W] J. Weidmann: Linear Operators in Hilbert Spaces. Springer-Verlag, New York-Heidelberg-Berlin 1980. MR 0566954 | Zbl 1025.47001
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