Asymptotic periodicity of solutions to a class of neutral functionaldifferential equations
Author:
Jian Hong Wu
Journal:
Proc. Amer. Math. Soc. 113 (1991), 355363
MSC:
Primary 34K15; Secondary 34K20
MathSciNet review:
1079900
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Abstract 
References 
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Abstract: In this paper, we extend a convergence result due to Takáč to continuous maps satisfying certain monotonicity properties. Applying this extension to the Poincaré map associated with the neutral equation we prove that each solution of the above neutral equation tends to an periodic function as in an oscillatory manner, where is an periodic continuous function and satisfies a certain order relation.
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 [1]
 O. Arino and E. Haourgui, On the asymptotic behavior of solutions of some delay differential systems which have a first integral, J. Math. Anal. Appl. 122 (1987), 3646. MR 874957 (88b:34115)
 [2]
 O. Arino and P. Seguier, About the behavior at infinity of solutions of , J. Math. Anal. Appl. 96 (1983), 420436. MR 719326 (84m:34107)
 [3]
 F. Atkinson and J. Haddock, Criteria for asymptotic constancy of solutions of functional differential equations, J. Math. Anal. Appl. 91 (1983), 410423. MR 690880 (84f:34096)
 [4]
 F. Atkinson, J. Haddock, and O. Staffans, Integral inequalities and exponential convergence of differential equations with bounded delay, Ordinary and Partial Differential Equations (W. Everitt and B. Sleeman, eds.), Lecture Notes in Math., SpringerVerlag, New York, 1982, pp. 5668. MR 693101 (84g:34123)
 [5]
 K. L. Cooke and J. L. Kaplan, A periodicity threshold theorem for epidemic and population growth, Math. Biosci. 31 (1976), 87104. MR 0682251 (58:33129)
 [6]
 K. L. Cooke and J. Yorke, Some equations modeling growth process and gonorrhea epidemics, Math. Biosci. 16 (1973), 75107. MR 0312923 (47:1478)
 [7]
 I. Györi, Connections between compartmental systems with pipes and integrodifferential equations, Math. Model. 7 (1986), 12151238. MR 877750 (89c:92020)
 [8]
 J. R. Haddock and J. Terjeki, LiapunovRazumikhin functions and invariance principle for functional differential equations, J. Differential Equations 48 (1983), 95122. MR 692846 (84f:34104)
 [9]
 J. R. Haddock, T. Krisztin, and J. Wu, Asymptotic equivalence of neutral equations and infinite delay equations, Nonlinear Anal. TMA 14 (1990), 369377. MR 1040012 (91b:34134)
 [10]
 J. K. Hale, Theory of functional differential equations, SpringerVerlag, New York, 1977. MR 0508721 (58:22904)
 [11]
 M. Hirsch, Attractors for discretetime monotone dynamical systems in strongly ordered spaces, Geometry and Topology, Lecture Notes in Math., vol. 1167, SpringerVerlag, Berlin, Heidelberg, and New York, 1985. MR 827267 (87e:58134)
 [12]
 M. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 153. MR 921986 (89c:58108)
 [13]
 J. K. Kaplan, M. Sorg, and J. Yorke, Solutions of have limits when is an order relation, Nonlinear Anal. TMA 3 (1979), 5358. MR 520471 (80b:34088)
 [14]
 H. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal. 17 (1986), 12891318. MR 860914 (87m:34057)
 [15]
 O. J. Staffans, A neutral FDE with stable operator is retarded, J. Differential Equations 49 (1983), 208217. MR 708643 (85j:34144)
 [16]
 P. Takáč, Convergence to equilibrium on invariant hypersurface for strongly increasing discretetime semigroup, preprint, 1989.
 [17]
 J. Wu, Convergence of monotone dynamical systems with minimal equilibria, Proc. Amer. Math. Soc. 106 (1989), 907911. MR 1004632 (90j:58130)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199110799002
PII:
S 00029939(1991)10799002
Keywords:
Asymptotic periodicity,
neutral equations,
monotone systems
Article copyright:
© Copyright 1991
American Mathematical Society
