Stability of Iterative Sequences Approximating Common Fixed Point for a System of Asymptotically Quasi-nonexpansive Type Mappings

  • Li, Jun (School of Mathematics and Information, China West Normal University) ;
  • Huang, Nan-Jing (Department of Mathematics, Sichuan University) ;
  • Cho, Yeol Je (Department of Mathematics Education and the RINS, Gyeongsang National University)
  • Received : 2005.11.30
  • Published : 2007.03.23

Abstract

In this paper, we introduce the concept of a system of asymptotically quasinonexpansive type mappings. Furthermore, we define a $k$-step iterative sequence approximating common fixed point for a system of asymptotically quasi-nonexpansive type mappings and study its stability in real Banach spaces.

Keywords

References

  1. R. P. Agarwal, N. J. Huang and Y. J. Cho, Stability of iterative procedures with errors for nonlinear equations of $\phi$-strongly accretive type operators, Numer. Funct. Anal. Optimiz., 22(2001), 471-485. https://doi.org/10.1081/NFA-100105303
  2. R. P. Agarwal, Y. J. Cho, J. Li and N. J. Huang, Stability of iterative procedures with errors approximating common fixed point for a couple of quasi-contractive mappings in q-uniformly smooth Banach spaces, J. Math. Anal. Appl., 272(2002), 435-447. https://doi.org/10.1016/S0022-247X(02)00150-6
  3. S. S. Chang, On the approximating problem of fixed points for asymptotically nonexpansive mappings, Indian J. Pure and Appl., 32(9)(2001), 1-11.
  4. M. K. Ghosh and L. Debnath, Convergence of Ishikawa iterative of quasi-nonexpansive mappings, J. Math. Anal. Appl., 207(1997), 96-103. https://doi.org/10.1006/jmaa.1997.5268
  5. K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35(1)(1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  6. A. M. Harder, Fixed Point Theory and Stability Results for Fixed Point Iteration Procedures, Ph.D. Thesis, University of Missouru-Rolla, (1987).
  7. A. M. Harder and T. L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon., 33(1988), 687-692.
  8. A. M. Harder and T. L. Hicks, Stability results for fixed point iteration procedures, Math. Japon., 33(1988), 693-706.
  9. Z.Y. Huang, Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl., 37(1999), 1-7.
  10. W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math., 17(1974), 339-346. https://doi.org/10.1007/BF02757136
  11. Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 259(2001), 1-7. https://doi.org/10.1006/jmaa.2000.6980
  12. Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl., 259(2001), 18-24. https://doi.org/10.1006/jmaa.2000.7353
  13. Q. H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex Banach spaces, J. Math. Anal. Appl., 266(2002), 468-471. https://doi.org/10.1006/jmaa.2001.7629
  14. M. O. Osilike, Stable iteration procedures for strong pseudocontractions and nonlinear equations of the accretive type, J. Math. Anal. Appl., 204(1996), 677-692. https://doi.org/10.1006/jmaa.1996.0461
  15. M. O. Osilike, Stability of the Mann and Ishikawa iteration procedures for $\phi$-strong pseudocontractions and nonlinear equations of the $\phi$-strongly accretive type, J. Math. Anal. Appl., 227(1998), 319-334. https://doi.org/10.1006/jmaa.1998.6075
  16. W. V. Petryshyn and T. E. Williamson, Strong and weak convergence of the sequence of successive approximations for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 43(1973), 459-497. https://doi.org/10.1016/0022-247X(73)90087-5
  17. B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure and Appl., 21(1990), 1-9.
  18. B. E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures II, Indian J. Pure and Appl., 24(1993), 691-703.
  19. J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 158(1991), 407-413. https://doi.org/10.1016/0022-247X(91)90245-U
  20. K. K. Tan and H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 122(3)(1994), 733-739. https://doi.org/10.1090/S0002-9939-1994-1203993-5
  21. K. K. Tan and H. K. Xu, Approximating fixed point of nonexpansive mappings by the Ishikawa iterative process, J. Math. Anal. Appl., 178(1993), 301-308. https://doi.org/10.1006/jmaa.1993.1309
  22. L. C. Zeng, A note on approximating fixed points of nonexpansive mapping by the Ishikawa iterative processes, J. Math. Anal. Appl., 226(1998), 245-250. https://doi.org/10.1006/jmaa.1998.6053