Strong Convergence of an explicit iteration method in uniformly convex Banach spaces

Abstract

We obtain the necessary and sufficient conditions for the convergence of an explicit iterative procedure to a common fixed point of a finite family of non-self asymptotically quasi-nonexpansive type mappings in real Banach spaces. We also prove the strong convergence of this iterative method to a common fixed point of a finite family of non-self asymptotically quasi-nonexpansive in the intermediate sense mappings in uniformly convex Banach spaces. Our results mainly generalize and extend those obtained by Wang [L. Wang, Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings, Computers \& Mathematics with applications, 53, (2007), 1012 - 1019.]

Keywords

• common fixed point,iterative approximation,asymptotically quasi-nonexpansive in the intermediate sense,uniformly convex Banach spaces

References

1. [1] S. S. Chang, Y. J. Cho, H. Zhou, Demiclosedness principle and weak convergence theorems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001), 1245-1260.
2. [2] C. E. Chidume, Nonexpansive mappings, Generalizations and iterative algorithms, Nonlinear Anal. Appl., vols. 1,2, Kluwer Academic, Dordrecht, 2003, pp. 383-421.
3. [3] C. E. Chidume, N. Shahzaad, strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal., 62(6) (2005), 1149-1156.
4. [4] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl.280 (2003), 364-374.
5. [5] K. Goebel, W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
6. [6] W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive semigroups, J. Math. Anal. Appl., 272 (2002), 565-574.
7. [7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
8. [8] S. Y. Matsuhita, D. Kuroiwa, Strong convergence of averaging iteration of nonexpansive nonself mappings, J. Math. Anal. Appl., 294 (2004), 206-214.

9. [9] M. O. Osilike, A. Udomene, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling , 32(2000), 1181-1191.
10. [10] M. O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl., 294 (2004), 73-81.
11. [11] S. Plubteing and R. Wangkeeree, Strong convergence theorems for three-step iterations with errors non-Lipschitzian nonself mappings in Banach spaces, Computers & Mathematics with applications, 51(2006), 1093-1102.
12. [12] S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67 (1979), 274-276.
13. [13] B. E. Rhoades, Fixed point iterations for certain nonlinear mappings, J. Math. Anal. Appl., 183(1994), 118-120.
14. [14] N. Shahzad, Approximating fixed points of nonself nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005), 1031-1039.
15. [15] J. schu, Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43(1991), 153-159.
16. [16] Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351-358.
17. [17] W. Takahashi, T.Tamura, convergence theorem for a pair of nonexpansive mappings, J. Conv. Anal., 5(1998), 45-56.
18. [18] K. K. Tan, H. K. Xu, The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Amer. Math. Soc. 114 (1992), 399-404.
19. [19] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.
20. [20] K. K. Tan, H. K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733-739.
21. [21] Y. X. Tian, S. S. Chang, J. L. Huang, On the approximation problem of common fixed points for a finite family of non-self asymptotically quasinonexpansive type mappings in Banach spaces, Computers & Mathematics with applications, 53(2007), 1847-1853.
22. [22] L. Wang, Strong and weak convergence theorems for nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323(2006), 550-557.
23. [23] L. Wang, Explicit iteration method for common fixed points of a finite family of nonself asymptotically nonexpansive mappings, Computers & Mathematics with applications, 53, (2007), Pages 1012 - 1019.
24. [24] H. K. Xu, X. M. Yin, Strong convergence theorems for nonexpansive nonself mappings, Nonlinear Anal. 24(2)(1995), 223-228.
25. [25] H. K. Xu, R. Ori, An implicit iterative process for nonexpansive mappings, Num. Funct. Anal. Optim. 22(2001), 767-773.
26. [26] H. Y. Zhou, S. S. Chang, Convergence of implicit iteration perocess for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. 23 (2002), 911-921.