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ON A PERTURBED CONSERVATIVE SYSTEM OF SEMILINEAR WAVE EQUATIONS WITH PERIODIC-DIRICHLET BOUNDARY CONDITIONS

Part of: Hyperbolic equations and systems Partial differential equations Partial differential operators

Published online by Cambridge University Press:  13 January 2010

JINHAI CHEN  and
DONAL O’REGAN
JINHAI CHEN*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Colorado Denver, Campus Box 170, PO Box 173364, Denver, CO 80217-3364, USA (email: [email protected])
DONAL O’REGAN
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
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Abstract

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In this paper, some existence and uniqueness results for generalized solutions to a periodic-Dirichlet problem for semilinear wave equations are given, using a global inverse function theorem. These results extend those known in the literature.

Keywords

wave equationperiodic-Dirichlet problemgeneralized solutionunique existenceglobal inverse function theorem

MSC classification

Secondary: 35B10: Periodic solutions 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations 47F05: Partial differential operators (should also be assigned at least one other classification number in section 47)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 281 - 288
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Bates, P. W., ‘Solutions of nonlinear elliptic systems with meshed spectra’, Nonlinear Anal. 4 (1980), 1023–1030.CrossRefGoogle Scholar
[2]Bates, P. W. and Castro, A., ‘Existence and uniqueness for a variational hyperbolic system without resonance’, Nonlinear Anal. 4 (1980), 1151–1156.CrossRefGoogle Scholar
[3]Dunford, N. and Schwartz, J. T., Linear Operators Part II: Spectral Theory, Self Adjoint Operators in Hilbert Space (Wiley Interscience, New York, 1988).Google Scholar
[4]Lazer, A. C., ‘Application of a lemma on bilinear forms to a problem in nonlinear oscillations’, Proc. Amer. Math. Soc. 33 (1972), 89–94.CrossRefGoogle Scholar
[5]Mawhin, J., ‘Contractive mappings and periodically perturbed conservative systems’, Arch. Math. (Brno) 12 (1976), 67–74.Google Scholar
[6]Mawhin, J., ‘Conservative systems of semilinear wave equations with periodic-Dirichlet boundary conditions’, J. Differential Equations 42 (1981), 116–128.CrossRefGoogle Scholar
[7]Mawhin, J. and Ward, J. R. Jr, ‘Asymptotic nonuniform nonresonance conditions in the periodic-Dirichlet problem for semilinear wave equations’, Ann. Mat. Pura Appl. 135 (1983), 85–97.CrossRefGoogle Scholar
[8]Plastock, R., ‘Homeomorphisms between Banach spaces’, Trans. Amer. Math. Soc. 200 (1974), 169–183.CrossRefGoogle Scholar
[9]Radulescu, M. and Radulescu, S., ‘Global inversion theorems and applications to differential equations’, Nonlinear Anal. 4 (1980), 951–965.CrossRefGoogle Scholar
[10]Shen, Z. H., ‘On the periodic solution to the Newtonian equation of motion’, Nonlinear Anal. 13 (1989), 145–149.Google Scholar
[11]Shen, Z. H. and Wolfe, M. A., ‘On the existence of periodic solutions of periodically perturbed conservative systems’, J. Math. Anal. Appl. 153 (1990), 78–83.Google Scholar