On solutions of a nonlinear wave question when the ratio of the period to the length of the interval is irrational
Author:
P. J. McKenna
Journal:
Proc. Amer. Math. Soc. 93 (1985), 5964
MSC:
Primary 35B10; Secondary 35L70
MathSciNet review:
766527
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Abstract: The semilinear wave equation with , is periodic in , is considered for some situations in which is not a rational multiple of . Various existence results depending on the range of are given, which contrast sharply with the case where is a rational multiple of .
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 A. Bahri and H. Brezis, Periodic solutions of a nonlinear wave equation, Proc. Roy. Soc. Edinburgh Sect. A 85 (1980), 313320. MR 574025 (82f:35011)
 [2]
 H. Brezis and L. Nirenberg, Characterization of the ranges of some nonlinear operators, and applications to boundary value problems, Ann. Scuola Norm. Pisa Cl. Sci. 5 (1978), 225326. MR 0513090 (58:23813)
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 J. Cassels, An introduction to Diophantine approximation, Cambridge Univ. Press, Cambridge, 1965. MR 0087708 (19:396h)
 [5]
 A. C. Lazer, Application of a lemma on bilinear forms to a problem in nonlinear oscillations, Proc. Amer. Math. Soc. 33 (1972), 8994. MR 0293179 (45:2258)
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 P. J. McKenna, On the reduction of a semilinear hyperbolic problem to a LandesmanLazer problem, Houston J. Math. 4 (1978), 577581. MR 523615 (83b:35106)
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 I. Niven and H. S. Zuckerman, The theory of numbers, 4th ed., Wiley, New York, 1980. MR 572268 (81g:10001)
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 P. H. Rabinowitz, Some minimax theorems and applications to nonlinear partial differential equations, Nonlinear Analysis, a volume in honor of E. H. Rothe, Academic Press, 1976. MR 0501092 (58:18545)
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 G. R. Sell, The prodigal integral, Amer. Math. Monthly 84 (1977), 162167. MR 0427556 (55:587)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002993919850766527X
PII:
S 00029939(1985)0766527X
Keywords:
Boundary value problems,
hyperbolic contractions,
compactness
Article copyright:
© Copyright 1985
American Mathematical Society
