Article
MSC:
35B10,
35G30,
45K05,
47A10,
73K12,
74K10 | MR 0695184 | Zbl 0533.35003 | DOI: 10.21136/AM.1983.104011
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Keywords:
periodic solution; nonlinear beam equation; existence
periodic solution; nonlinear beam equation; existence
Summary:
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 $f\in C\left(\left[0,\omega\right],L_2\right)$.
References:
[1] V. B. Litvinov, Ju. V. Jadykin: Vibration of Extensible Thin Bodies in Transversal Flow. Dokl. Akad. Nauk Ukrain. SSR, Ser. A, No 4 (1981) 47-50.
[2] J. M. Ball: Stability Theory for an Extensible Beam. J. Differential Equations 14 (1973), 399-418. DOI 10.1016/0022-0396(73)90056-9 | MR 0331921 | Zbl 0247.73054
[3] T. Narazaki: On the Time Global Solutions of Perturbed Beam Equations. Proc. Fac. Sci. Tokai Univ. 16 (1981), 51-71. MR 0632661 | Zbl 0474.35010
[4] V. Lovicar: Periodic Solutions of Nonlinear Abstract Second Order Equations with Dissipative Terms. Čas. Pěst. Mat. 102 (1977), 364-369. MR 0508656 | Zbl 0369.34017
[5] N. Dunford J. T. Schwartz: Linear operators I. (Intersci. Publ. New York-London) 1958. MR 0117523
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