# On periodic solution of a nonlinear beam equation

Aplikace matematiky (1983)

• Volume: 28, Issue: 2, page 108-115
• ISSN: 0862-7940

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## Abstract

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

## How to cite

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Kopáčková, Marie. "On periodic solution of a nonlinear beam equation." Aplikace matematiky 28.2 (1983): 108-115. <http://eudml.org/doc/15282>.

@article{Kopáčková1983,
abstract = {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 $f\in C\left(\left[0,\omega \right],L_2\right)$.},
author = {Kopáčková, Marie},
journal = {Aplikace matematiky},
keywords = {periodic solution; nonlinear beam equation; existence; periodic solution; nonlinear beam equation; existence},
language = {eng},
number = {2},
pages = {108-115},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On periodic solution of a nonlinear beam equation},
url = {http://eudml.org/doc/15282},
volume = {28},
year = {1983},
}

TY - JOUR
AU - Kopáčková, Marie
TI - On periodic solution of a nonlinear beam equation
JO - Aplikace matematiky
PY - 1983
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 28
IS - 2
SP - 108
EP - 115
AB - 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 $f\in C\left(\left[0,\omega \right],L_2\right)$.
LA - eng
KW - periodic solution; nonlinear beam equation; existence; periodic solution; nonlinear beam equation; existence
UR - http://eudml.org/doc/15282
ER -

## References

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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. (1981)
2. J. M. Ball, 10.1016/0022-0396(73)90056-9, J. Differential Equations 14 (1973), 399-418. (1973) Zbl0247.73054MR0331921DOI10.1016/0022-0396(73)90056-9
3. T. Narazaki, On the Time Global Solutions of Perturbed Beam Equations, Proc. Fac. Sci. Tokai Univ. 16 (1981), 51-71. (1981) Zbl0474.35010MR0632661
4. V. Lovicar, Periodic Solutions of Nonlinear Abstract Second Order Equations with Dissipative Terms, Čas. Pěst. Mat. 102 (1977), 364-369. (1977) Zbl0369.34017MR0508656
5. N. Dunford J. T. Schwartz, Linear operators I, (Intersci. Publ. New York-London) 1958. (1958) Zbl0084.10402MR0117523

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