## Topological Methods in Nonlinear Analysis

### Forced oscillations in strongly damped beam equation

Aleksander Ćwiszewski

#### Abstract

It is proved that the extensible beam equation in Ball's model admits periodic solutions near equilibrium states if subject to external periodic force of high frequency. The approach is based on translation along trajectories, averaging method and homotopy invariants such as topological degree and fixed point index.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 37, Number 2 (2011), 259-282.

Dates
First available in Project Euclid: 20 April 2016

https://projecteuclid.org/euclid.tmna/1461184786

Mathematical Reviews number (MathSciNet)
MR2849823

Zentralblatt MATH identifier
1248.47059

#### Citation

Ćwiszewski, Aleksander. Forced oscillations in strongly damped beam equation. Topol. Methods Nonlinear Anal. 37 (2011), no. 2, 259--282. https://projecteuclid.org/euclid.tmna/1461184786

#### References

• \ref\no\dfazeroR. R. Akhmerov, M. I. Kamenskiĭ. A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Birkhäuser (1992)
• \ref\no\dfaBall J. Ball, Stability theory for an extensible beam , J. Differential Equations, 14 (1973), 399–418 \ref\no\dfaBall1 ––––, Initial-boundary value problems for an extensible beam, J. Math. Anal. Appl., 42 (1973), 61–90
• \ref\no \dfaCholewa-Dlotko J. W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press (2000) \ref\no\dfaCo
• C. C. Conley, Isolated Invariant Sets and Isolating Block, CBMS, 38 , Amer. Math. Soc. (1978) \ref\no\dfaCwiszewski-JDE2009
• A. Ćwiszewski, Topological degree methods for perturbations of operators generating compact $C_0$ semigroups , J. Differential Equations, 220 \rom(2) (2006), 434–477 \ref\no\dfaCwiszewski-CEJM ––––, Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach , Cent. Eur. J. Math., 9 \rom(2) (2011), 244–268 \ref\no\dfaCwiszewski-Kokocki-DCDSA
• A. Ćwiszewski and P. Kokocki, Krasnosel'skiĭ type formula and translation along trajectories method for evolution equations, , Discrete Contin. Dynam. Systems Ser. A, 22 (2008), 605–628 \ref\no\dfaCwiszewski-Rybakowski
• A. Ćwiszewski and K. P. Rybakowski, Singular dynamics of strongly damped beam equation , J. Differential Equations, 247 (2009), 3202–3233 \ref\no\dfaDeimling
• K. Deimling, Multivalued Differential Equations, Walter de Gruyter (2000)
• \ref\no \dfaFitzgibbon W. E. Fitzgibbon, Strongly damped quasilinear evolution equations , J. Math. Anal. Appl., 79 \rom(2) (1981), 536–550
• \ref\no\dfaHenry-book D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer–Verlag (1981) \ref\no\dfaMassat
• P. Massatt, Limiting behavior for strongly damped nonlinear wave equations , J. Differential Equations, 48 (1983), 334–349 \ref\no\dfaNussbaum
• R. D. Nussbaum, The fixed point index for local condensing maps , Ann. Mat. Pura Appl., 89 (1971), 217–258 \ref\no \dfaSevcovic
• D. Ševčovič, Limiting behavior of global attractors for singularly perturbed beam equations with strong damping , Comment. Math. Univ. Carolinae, 32 \rom(1) (1991), 45–60