Path stability and nonlinear weak ergodic theorems

Chen, Yong-Zhuo

doi:10.1090/S0002-9947-00-02600-3

Path stability and nonlinear weak ergodic theorems
HTML articles powered by AMS MathViewer

by Yong-Zhuo Chen PDF

Trans. Amer. Math. Soc. 352 (2000), 5279-5292 Request permission

Abstract:

Let $\{f_{n} \}$ be a sequence of nonlinear operators. We discuss the asymptotic properties of their inhomogeneous iterates $f_{n} \circ f_{n-1} \circ \cdots \circ f_{1}$ in metric spaces, then apply the results to the ordered Banach spaces through projective metrics. Theorems on path stability and nonlinear weak ergodicity are obtained in this paper.

References

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. MR 239559, DOI 10.1090/S0002-9939-1969-0239559-9
T. A. Burton, Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2383–2390. MR 1346965, DOI 10.1090/S0002-9939-96-03533-2
Yong Zhuo Chen, Thompson’s metric and mixed monotone operators, J. Math. Anal. Appl. 177 (1993), no. 1, 31–37. MR 1224803, DOI 10.1006/jmaa.1993.1241
Y.-Z. Chen, Inhomogeneous iterates of contraction mappings and nonlinear ergodic theorems, Nonlinear Analysis 39(2000), 1–10.
Da Jun Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, vol. 5, Academic Press, Inc., Boston, MA, 1988. MR 959889
Takao Fujimoto and Ulrich Krause, Asymptotic properties for inhomogeneous iterations of nonlinear operators, SIAM J. Math. Anal. 19 (1988), no. 4, 841–853. MR 946646, DOI 10.1137/0519058
Takao Fujimoto and Ulrich Krause, Stable inhomogeneous iterations of nonlinear positive operators on Banach spaces, SIAM J. Math. Anal. 25 (1994), no. 4, 1195–1202. MR 1278900, DOI 10.1137/S0036141091221651
Hisashi Inaba, Weak ergodicity of population evolution processes, Math. Biosci. 96 (1989), no. 2, 195–219. MR 1015666, DOI 10.1016/0025-5564(89)90059-X
Jacek R. Jachymski, An extension of A. Ostrowski’s theorem on the round-off stability of iterations, Aequationes Math. 53 (1997), no. 3, 242–253. MR 1444177, DOI 10.1007/BF02215974
James P. Keener, The Perron-Frobenius theorem and the ranking of football teams, SIAM Rev. 35 (1993), no. 1, 80–93. MR 1207798, DOI 10.1137/1035004
M. A. Krasnosel′skiĭ and P. P. Zabreĭko, Geometrical methods of nonlinear analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 263, Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. MR 736839, DOI 10.1007/978-3-642-69409-7
U. Krause, Path stability of prices in a nonlinear Leontief model, Ann. Oper. Res. 37(1992), 141-148.
U. Krause, Positive nonlinear systems: some results and applications, Proceedings of the First World Congress of Nonlinear Analysts 1992, W. de Gruyter, Berlin, 1996.
Roger D. Nussbaum, Hilbert’s projective metric and iterated nonlinear maps, Mem. Amer. Math. Soc. 75 (1988), no. 391, iv+137. MR 961211, DOI 10.1090/memo/0391
Roger D. Nussbaum, Some nonlinear weak ergodic theorems, SIAM J. Math. Anal. 21 (1990), no. 2, 436–460. MR 1038901, DOI 10.1137/0521024
E. Seneta, Nonnegative matrices and Markov chains, 2nd ed., Springer Series in Statistics, Springer-Verlag, New York, 1981. MR 719544, DOI 10.1007/0-387-32792-4
A. C. Thompson, On certain contraction mappings in a partially ordered vector space. , Proc. Amer. Math. soc. 14 (1963), 438–443. MR 0149237, DOI 10.1090/S0002-9939-1963-0149237-7