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periodic solution; nonlinear beam equation; existence
the existence of an $\omega$-periodic solution of the equation $\frac {\partial^2u}{\partial t^2} + \alpha \frac {\partial^4u} {\partial x^4} + \gamma \frac {\partial^5u}{\partial x^4\partial t} - \tilde{\gamma} \frac {\partial^3u}{\partial x^2\partial t} + \delta \frac {\partial u}{\partial t} - \left[\beta + \aleph\int^n_0{\left(\frac {\partial u}{\partial x}\right)}^2 (\cdot,\xi)d\xi + \sigma \int^n_0 \frac {\partial^2u}{\partial x \partial t} (\cdot,\xi) \frac {\partial u}{\partial x}(\cdot,\xi)d \xi \right] \frac {\partial^2u}{\partial x^2}=f$ sarisfying the boundary conditions $u(t,0)=u(t,\pi)=\frac{\partial^2u}{\partial x^2}\left(t,0\right)=\frac{\partial^2u}{\partial x^2}\left(t,\pi\right)=0$ is proved for every $\omega$-periodic function $f\in C\left(\left[0,\omega\right],L_2\right)$.
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