##
**Periodic solutions of nonlinear vibrating beams.**
*(English)*
Zbl 1074.35084

Authors investigate the existence and multiplicity of solutions for periodic semilinear beam equation problems of the form
\[
\begin{aligned} \partial_{t}^{2} u + \alpha_{0}^{2}\partial_{x}^{4} u - g(x,u) &= h(x,t),\\ u(0,t) = u(L,t) = \partial_{x}^{2} u(0,t) &= \partial_{x}^{2} u(L,t) = 0\qquad (x\in]0,L[,t\in \mathbb R),\\ u(x,t) &= u(x,t+T), \end{aligned}
\]
where \(h\) is \(T\)-periodic in time, \(\alpha_{0} > 0\) is a constant. Moreover, \(g\) is time independent and thus the period \(T\) of periodic solutions is determined only by the forcing term \(h\). The authors obtain sufficient conditions for the time-independent nonlinear term \(g(x,u)\) and the period \(T\) such that there is a weak solution of the problem for any \(T\)-periodic forcing term \(h\). As for the multiplicity of solutions, the authors show that multiple solutions exist for any small forcing term \(h\) with suitable period \(T\) under some conditions on the interaction between the nonlinearity and the spectrum of the beam operator. The main tools are the Banach fixed-point theorem and the generalized Leray-Schauder degree introduced by J. Berkovits and V. Mustonen [Differ. Integral Equ. 3, No. 5, 945–963 (1990; Zbl 0724.47024)] together with homotopy arguments. Finally, let me note that the whole paper is really excellently organized and the obtained results are illustrated by several examples.

Reviewer: Petr Necesal (Plzen)

### MSC:

35Q72 | Other PDE from mechanics (MSC2000) |

35B10 | Periodic solutions to PDEs |

35L35 | Initial-boundary value problems for higher-order hyperbolic equations |

47H11 | Degree theory for nonlinear operators |

47N20 | Applications of operator theory to differential and integral equations |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74H45 | Vibrations in dynamical problems in solid mechanics |