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Nonlinear oscillations in a suspension bridge. (English) Zbl 0676.35003

This paper deals with a simplified mathematical model for a suspension bridge \[ u_{tt}+K_1u_{xxxx} + K_2u^+ = W(x) + \varepsilon f(x,t);\quad u(0,t)=u(L,t)=0,\quad u_{xx}(0,t)=u_{xx}(L,t)=0, \tag{1} \] where \(W(x)\) is the weight per unit length at \(x\), and \(f(x,t)\) is an externally imposed periodic function.
A. C. Lazer and the first author [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 243–274 (1987; Zbl 0633.34037)] considered (1) under the assumption that \(u(x) = W_0 \sin (\pi x/L)\).
The paper in question deals with periodic solution of (1) in the more realistic case that \(W(x)=W_0\), \(0<x<L\).
Reviewer: Jinghuang Tian

MSC:

35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35G15 Boundary value problems for linear higher-order PDEs
35B10 Periodic solutions to PDEs

Citations:

Zbl 0633.34037
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References:

[1] Coron, J. M., Periodic solutions of a nonlinear wave equation without assumptions of monotonicity. Math. Ann. 262 (1983), 273–285. · Zbl 0489.35061 · doi:10.1007/BF01455317
[2] Lazer, A. C, &amp; Mckenna, P. J., Large scale oscillatory behaviour in loaded asymmetric systems. Annales Inst. H. Poincaré, to appear. · Zbl 0633.34037
[3] Lazer, A. C., &amp; McKenna, P. J., A symmetry theorem and applications to nonlinear partial differential equations, to appear. · Zbl 0666.47038
[4] McKenna, P. J., &amp; Walter, W., On the multiplicity of the solution set of some nonlinear boundary value problems. Nonlinear Analysis 8 (1984), 893–907. · Zbl 0556.35024 · doi:10.1016/0362-546X(84)90110-X
[5] Nirenberg, L., Topics in Nonlinear Functional Analysis. Courant Inst. Lecture Notes (1974). · Zbl 0286.47037
[6] Schröder, J., Operator Inequalities. Academic Press 1980.
[7] Vejvoda, O., Partial Differential Equations. Nordhoff, Sigthoff 1981. · Zbl 0458.47046
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