Geodesic
Small time-periodic solutions to a nonlinear equation of a vibrating string
Applications of Mathematics, Tome 32 (1987) no. 6, pp. 480-490
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In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.
In this paper, the system consisting of two nonlinear equations is studied. The former is hyperbolic with a dissipative term and the latter is elliptic. In a special case, the system reduces to the approximate model for the damped transversal vibrations of a string proposed by G. F. Carrier and R. Narasimha. Taking advantage of accelerated convergence methods, the existence of at least one time-periodic solution is stated on condition that the right-hand side of the system is sufficiently small.
DOI : 10.21136/AM.1987.104278
Classification : 35B10, 35L70, 58C15, 73K03
Keywords: nonlinear string equation; accelerated convergence; existence; periodic; Dirichlet boundary conditions; vibrations; damped extensive string; time-periodic solution 
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Feireisl, Eduard. Small time-periodic solutions to a nonlinear equation of a vibrating string. Applications of Mathematics, Tome 32 (1987) no. 6, pp. 480-490. doi: 10.21136/AM.1987.104278

[1] R. W. Dickey: Infinite systems of nonlinear oscillation equations with Linear Damping. Siam J. Appl. Math. 19 (1970), pp. 208-214. | DOI | MR | Zbl

[2] S. Klainerman: Global existence for nonlinear wave equations. Comm. Pure Appl. Math. 33 (1980), pp. 43--101. | DOI | MR | Zbl

[3] P. Krejčí: Hard Implicit Function Theorem and Small periodic solutions to partial differential equations. Comment. Math. Univ. Carolinae 25 (1984), pp. 519-536. | MR

[4] J. Moser: A rapidly-convergent iteration method and nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa 20-3 (1966), pp. 265-315, 499-535.

[5] O. Vejvoda, et al.: Partial differential equations: Time-periodic Solutions. Martinus Nijhoff Publ., 1982. | Zbl

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