## 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

### Keywords:

nonuniqueness; time-periodical solutions
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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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