Convergence in almost periodic cooperative systems with a first integral

Shen, Wenxian; Zhao, Xiao-Qiang

doi:10.1090/S0002-9939-04-07556-2

Convergence in almost periodic cooperative systems with a first integral
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by Wenxian Shen and Xiao-Qiang Zhao PDF

Proc. Amer. Math. Soc. 133 (2005), 203-212 Request permission

Abstract:

This paper is to investigate the asymptotic dynamics in almost periodic cooperative systems with a first integral. By appealing to the theory of skew-product semiflows we establish the asymptotic almost periodicity of bounded solutions to such systems, which extends the existing convergence results for time independent and periodic cooperative systems with a first integral and proves a conjecture of B. Tang, Y. Kuang and H. Smith in SIAM J. Math. Anal., 24 (1993), 1331-1339.

References

Ovide Arino, Monotone semi-flows which have a monotone first integral, Delay differential equations and dynamical systems (Claremont, CA, 1990) Lecture Notes in Math., vol. 1475, Springer, Berlin, 1991, pp. 64–75. MR 1132019, DOI 10.1007/BFb0083480
Xu-Yan Chen and Hiroshi Matano, Convergence, asymptotic periodicity, and finite-point blow-up in one-dimensional semilinear heat equations, J. Differential Equations 78 (1989), no. 1, 160–190. MR 986159, DOI 10.1016/0022-0396(89)90081-8
A. M. Fink, Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974. MR 0460799, DOI 10.1007/BFb0070324
Georg Hetzer and Wenxian Shen, Uniform persistence, coexistence, and extinction in almost periodic/nonautonomous competition diffusion systems, SIAM J. Math. Anal. 34 (2002), no. 1, 204–227. MR 1950832, DOI 10.1137/S0036141001390695
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
Jifa Jiang, Type $K$-monotone systems with an order-increasing invariant function, Chinese Ann. Math. Ser. B 17 (1996), no. 3, 335–342. A Chinese summary appears in Chinese Ann. Math. Ser. A 17 (1996), no. 4, 516. MR 1415261
Ji-Fa Jiang, Periodic monotone systems with an invariant function, SIAM J. Math. Anal. 27 (1996), no. 6, 1738–1744. MR 1416516, DOI 10.1137/S003614109326063X
Janusz Mierczyński, Strictly cooperative systems with a first integral, SIAM J. Math. Anal. 18 (1987), no. 3, 642–646. MR 883558, DOI 10.1137/0518049
F. Nakajima, Periodic time dependent gross-substitute systems, SIAM J. Appl. Math. 36 (1979), no. 3, 421–427. MR 531605, DOI 10.1137/0136032
Peter Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), no. 1, 89–110. MR 997611, DOI 10.1016/0022-0396(89)90115-0
Robert J. Sacker and George R. Sell, Lifting properties in skew-product flows with applications to differential equations, Mem. Amer. Math. Soc. 11 (1977), no. 190, iv+67. MR 448325, DOI 10.1090/memo/0190
George R. Sell, Topological dynamics and ordinary differential equations, Van Nostrand Reinhold Mathematical Studies, No. 33, Van Nostrand Reinhold Co., London, 1971. MR 0442908
George R. Sell and Fumio Nakajima, Almost periodic gross-substitute dynamical systems, Tohoku Math. J. (2) 32 (1980), no. 2, 255–263. MR 580280, DOI 10.2748/tmj/1178229641
Wenxian Shen and Yingfei Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc. 136 (1998), no. 647, x+93. MR 1445493, DOI 10.1090/memo/0647
Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
Peter Takáč, Asymptotic behavior of strongly monotone time-periodic dynamical processes with symmetry, J. Differential Equations 100 (1992), no. 2, 355–378. MR 1194815, DOI 10.1016/0022-0396(92)90119-8
Bao Rong Tang, Yang Kuang, and Hal Smith, Strictly nonautonomous cooperative system with a first integral, SIAM J. Math. Anal. 24 (1993), no. 5, 1331–1339. MR 1234019, DOI 10.1137/0524076
X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494–509.
X.-Q. Zhao, “Dynamical Systems in Population Biology”, CMS books in Mathematics, Vol. 16, Springer-Verlag, New York, 2003.