Asymptotic periodicity of solutions to a class of neutral functional-differential equations
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- by Jian Hong Wu
- Proc. Amer. Math. Soc. 113 (1991), 355-363
- DOI: https://doi.org/10.1090/S0002-9939-1991-1079900-2
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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 \[ (d/dt)[x(t) - b(t)x(t - r)] = F[t,x(t),x(t - r)]\] we prove that each solution of the above neutral equation tends to an $r$-periodic function as $t \to \infty$ in an oscillatory manner, where $0 \leq b(t) < 1$ is an $r$-periodic continuous function and $F$ satisfies a certain order relation.References
- O. Arino and E. Haourigui, On the asymptotic behavior of solutions of some delay differential systems which have a first integral, J. Math. Anal. Appl. 122 (1987), no. 1, 36–46. MR 874957, DOI 10.1016/0022-247X(87)90342-8
- O. Arino and P. Séguier, About the behaviour at infinity of solutions of $x^{\prime } (t)=f(t-1,\,x(t-1))-f(t,\,x(t))$, J. Math. Anal. Appl. 96 (1983), no. 2, 420–436. MR 719326, DOI 10.1016/0022-247X(83)90051-3
- F. V. Atkinson and J. R. Haddock, Criteria for asymptotic constancy of solutions of functional-differential equations, J. Math. Anal. Appl. 91 (1983), no. 2, 410–423. MR 690880, DOI 10.1016/0022-247X(83)90161-0
- F. V. Atkinson, J. R. Haddock, and O. J. Staffans, Integral inequalities and exponential convergence of solutions of differential equations with bounded delay, Ordinary and partial differential equations (Dundee, 1982) Lecture Notes in Math., vol. 964, Springer, Berlin-New York, 1982, pp. 56–68. MR 693101
- Kenneth L. Cooke and James L. Kaplan, A periodicity threshold theorem for epidemics and population growth, Math. Biosci. 31 (1976), no. 1-2, 87–104. MR 682251, DOI 10.1016/0025-5564(76)90042-0
- Kenneth L. Cooke and James A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16 (1973), 75–101. MR 312923, DOI 10.1016/0025-5564(73)90046-1
- István Győri, Connections between compartmental systems with pipes and integro-differential equations, Math. Modelling 7 (1986), no. 9-12, 1215–1238. Mathematical models in medicine: diseases and epidemics, Part 2. MR 877750, DOI 10.1016/0270-0255(86)90077-1
- J. R. Haddock and J. Terjéki, Liapunov-Razumikhin functions and an invariance principle for functional-differential equations, J. Differential Equations 48 (1983), no. 1, 95–122. MR 692846, DOI 10.1016/0022-0396(83)90061-X
- J. R. Haddock, T. Krisztin, and Jian Hong Wu, Asymptotic equivalence of neutral and infinite retarded differential equations, Nonlinear Anal. 14 (1990), no. 4, 369–377. MR 1040012, DOI 10.1016/0362-546X(90)90171-C
- Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721, DOI 10.1007/978-1-4612-9892-2
- Morris W. Hirsch, Attractors for discrete-time monotone dynamical systems in strongly ordered spaces, Geometry and topology (College Park, Md., 1983/84) Lecture Notes in Math., vol. 1167, Springer, Berlin, 1985, pp. 141–153. MR 827267, DOI 10.1007/BFb0075221
- Morris W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. Reine Angew. Math. 383 (1988), 1–53. MR 921986, DOI 10.1515/crll.1988.383.1
- James L. Kaplan, M. Sorg, and James A. Yorke, Solutions of $x^{\prime } (t)=f(x(t),\,x(t-L))$ have limits when $f$ is an order relation, Nonlinear Anal. 3 (1979), no. 1, 53–58 (1978). MR 520471, DOI 10.1016/0362-546X(79)90034-8
- Hal L. Smith, Periodic solutions of periodic competitive and cooperative systems, SIAM J. Math. Anal. 17 (1986), no. 6, 1289–1318. MR 860914, DOI 10.1137/0517091
- Olof J. Staffans, A neutral FDE with stable $D$-operator is retarded, J. Differential Equations 49 (1983), no. 2, 208–217. MR 708643, DOI 10.1016/0022-0396(83)90012-8 P. Takáč, Convergence to equilibrium on invariant $d$-hypersurface for strongly increasing discrete-time semigroup, preprint, 1989.
- Jian Hong Wu, Convergence of monotone dynamical systems with minimal equilibria, Proc. Amer. Math. Soc. 106 (1989), no. 4, 907–911. MR 1004632, DOI 10.1090/S0002-9939-1989-1004632-7
Bibliographic Information
- © Copyright 1991 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 113 (1991), 355-363
- MSC: Primary 34K15; Secondary 34K20
- DOI: https://doi.org/10.1090/S0002-9939-1991-1079900-2
- MathSciNet review: 1079900