COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS BY A MODIFIED FASTER ITERATION SCHEME

- Journal title : Bulletin of the Korean Mathematical Society
- Volume 47, Issue 5, 2010, pp.973-985
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/BKMS.2010.47.5.973

Title & Authors

COMMON FIXED POINTS OF TWO NONEXPANSIVE MAPPINGS BY A MODIFIED FASTER ITERATION SCHEME

Khan, Safeer Hussain; Kim, Jong-Kyu;

Khan, Safeer Hussain; Kim, Jong-Kyu;

Abstract

We introduce an iteration scheme for approximating common fixed points of two mappings. On one hand, it extends a scheme due to Agarwal et al. [2] to the case of two mappings while on the other hand, it is faster than both the Ishikawa type scheme and the one studied by Yao and Chen [18] for the purpose in some sense. Using this scheme, we prove some weak and strong convergence results for approximating common fixed points of two nonexpansive self mappings. We also outline the proofs of these results to the case of nonexpansive nonself mappings.

Keywords

iteration scheme;nonexpansive self mapping;nonexpansive nonself mapping;rate of convergence;common fixed point;the condition (A`);weak and strong convergence;

Language

English

Cited by

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References

1.

M. Abbas, S. H. Khan, and J. K. Kim, A new one-step iterative process for common fixed points in Banach spaces, J. Inequal. Appl. 2008 (2008), Art. ID 548627, 10 pp.

2.

R. P. Agarwal, D. O’Regan, and D. R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007), no. 1, 61-79.

3.

F. E. Browder, Nonlinear operators and nolinear equations of evolution in Banach spaces, Proc. Symp. Pure Math., Vol. 18, Proc. Amer. Math. Soc., Providence, RI, 1976.

4.

R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math. 32 (1979), no. 2-3, 107-116.

5.

G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986), no. 11, 1263-1269.

6.

H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl. 328 (2007), no. 2, 821-829.

7.

W. Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, J. Math. Anal. Appl. 272 (2002), no. 2, 565-574.

8.

W. Kaczor and S. Prus, Asymptotical smoothness and its applications, Bull. Austral. Math. Soc. 66 (2002), no. 3, 405-418.

9.

S. H. Khan and H. Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Anal. 61 (2005), no. 8, 1295-1301.

10.

S. H. Khan and W. Takahashi, Approximating common fixed points of two asymptotically nonexpansive mappings, Sci. Math. Jpn. 53 (2001), no. 1, 143-148.

11.

G. Li and J. K. Kim, Demiclosedness principle and asymptotic behavior for nonexpansive mappings in metric spaces, Appl. Math. Lett. 14 (2001), no. 5, 645-649.

13.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

14.

J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), no. 1, 153-159.

15.

N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61 (2005), no. 6, 1031-1039.

16.

W. Takahashi and G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon. 48 (1998), no. 1, 1-9.

17.

W. Takahashi and T. Tamura, Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1998), no. 1, 45-56.

18.

Y. Yao and R. Chen, Weak and strong convergence of a modified Mann iteration for asymptotically nonexpansive mappings, Nonlinear Funct. Anal. Appl. 12 (2007), no. 2, 307-315.