Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Path stability and nonlinear weak ergodic theorems


Author: Yong-Zhuo Chen
Journal: Trans. Amer. Math. Soc. 352 (2000), 5279-5292
MSC (2000): Primary 47H07, 47H09; Secondary 47H10
DOI: https://doi.org/10.1090/S0002-9947-00-02600-3
Published electronically: July 12, 2000
MathSciNet review: 1707493
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969), 458-464. MR 39:916
  • 2. T. A. Burton, Integral equations, implicit functions and fixed points, Proc. Amer. Math. Soc. 124(1996), 2383-2390. MR 96j:45001
  • 3. Y.-Z. Chen, Thompson's metric and mixed monotone operators, J. Math. Anal. Appl. 117(1993), 31-37. MR 94d:47055
  • 4. Y.-Z. Chen, Inhomogeneous iterates of contraction mappings and nonlinear ergodic theorems, Nonlinear Analysis 39(2000), 1-10. CMP 2000:03
  • 5. D. Guo and V. Lakshimikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, 1988. MR 89k:47084
  • 6. T. Fujimoto and U. Krause, Asymptotic properties for inhomogeneous iterations of nonlinear operators, SIAM J. Math. Anal. 19(1988), 841-853. MR 90b:47119
  • 7. T. Fujimoto and U. Krause, Stable inhomogeneous iterations of nonlinear positive operators on Banach spaces, SIAM J. Math. Anal. 25(1994), 1195-1202. MR 95m:47104
  • 8. H. Inaba, Weak ergodicity of population evolution processes, Math. Biosc. 96(1989), 195-219. MR 91b:92028
  • 9. J. R. Jachymski, An extension of A. Ostrowski's Theorem on the round-off stability of iterations, Aequ. Math. 53(1997), 242-253. MR 98d:47122
  • 10. J. P. Keener, The Perron-Frobenius Theorem and the ranking of football teams, SIAM Review 35(1993), 80-93. MR 94a:15012
  • 11. M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984. MR 85b:47057
  • 12. U. Krause, Path stability of prices in a nonlinear Leontief model, Ann. Oper. Res. 37(1992), 141-148. CMP 93:01
  • 13. U. Krause, Positive nonlinear systems: some results and applications, Proceedings of the First World Congress of Nonlinear Analysts 1992, W. de Gruyter, Berlin, 1996. CMP 96:12
  • 14. R. D. Nussbaum, Iterated nonlinear maps and Hilbert's projective metric, Mem. Amer. Math. Soc. Vol.75, No.391 (Sept. 1988). MR 89m:47046
  • 15. R. D. Nussbaum, Some nonlinear weak ergodic theorems, SIAM. J. Math. Anal. 21(1990), pp. 436-460. MR 90m:47081
  • 16. E. Seneta, Non-negative Matrices and Markov Chains, 2nd ed., Springer-Verlag, Berlin, 1980; 1st ed., Non-negative Matrice, G. Allen and Unwin, London, 1973. MR 85i:60058
  • 17. A. C. Thompson, On certain contraction mappings in a partially ordered vector space, Proc. Amer. Math. Soc., 14(1963), 438-443. MR 26:6727

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 47H07, 47H09, 47H10

Retrieve articles in all journals with MSC (2000): 47H07, 47H09, 47H10


Additional Information

Yong-Zhuo Chen
Affiliation: Division of Natural Sciences, University of Pittsburgh at Bradford, Bradford, Pennsylvania 16701
Email: yong@imap.pitt.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02600-3
Keywords: Hilbert metric, inhomogeneous iterates, metric space, monotone operator, ordered Banach space, Thompson's metric
Received by editor(s): June 30, 1998
Received by editor(s) in revised form: June 1, 1999
Published electronically: July 12, 2000
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society