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On the existence of free vibrations for a beam equation when the period is an irrational multiple of the length. (English) Zbl 0684.35057

The author studies the existence of time-periodical solutions with period \(T>0\), u(x,t) of the equation \[ u_{tt}+i_{xxxx}+f(x,u)=0 \] with the boundary conditions: \[ u(0,t)=u(\pi,t)=0;\quad u_{xx}(0,t)=u_{xx}(\pi,t)=0, \] under suitable conditions for f(x,u).
The main goal is the proof of the theorem that for an arbitrary positive integer n there exists a real constant \(T_ 0>0\) such that for almost every \(T\in (T_ 0,\infty)\) (in the sense of the Lebesgue measure on \(R^ 1)\) there exists n different nontrivial solutions of the problem which are not translation of one another.
Reviewer: S.Nocilla

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35B10 Periodic solutions to PDEs
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References:

[1] J. M. Coron: Periodic solutions of a nonlinear wave equation without assumption of monotonicity. Math. Ann. 262 (1983), 273-285. · Zbl 0489.35061
[2] D. G. Costa M. Willem: Multiple critical points of invariant functional and applications. Séminaire de Mathématique 2-éme Semestre Université Catholique de Louvain. · Zbl 0628.35037
[3] I. Ekeland R. Temam: Convex analysis and variational problems. North-Holland Publishing Company 1976. · Zbl 0322.90046
[4] N. Krylová O. Vejvoda: A linear and weakly nonlinear equation of a beam: the boundary value problem for free extremities and its periodic solutions. Czechoslovak Math. J. 21 (1971), 535-566. · Zbl 0226.35008
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