Korean J. Math.  Vol 22, No 2 (2014)  pp.235-252
DOI: https://doi.org/10.11568/kjm.2014.22.2.235

Weak and strong convergence of three step iteration scheme with errors for non-self asymptotically nonexpansive mapping

Jae Ug Jeong, Young Chel Kwun


In this paper, weak and strong convergence theorems of three step iteration process with errors are established for two weakly inward and non-self asymptotically nonexpansive mappings in Banach spaces. the results obtained in this paper extend and improve the several recent results in this area.


Asymptotically nonexpansive mapping, Uniformly convex, Common fixed point; Three step iteration.

Subject classification

47H09; 47H10


This work was supported by Dong-eui university foundation grant (2014).

Full Text:



E. blum and W. Oettli, From optization and variational inequalities to equilib- rium problems, Math. Student 63 (1994), 123–145. (Google Scholar)

C. Byrne, A unified treatment of some iterative algorithms in signal processing and imagine reconstruction, Inverse Problems 20 (2004), 103–120. (Google Scholar)

S. S. Chang, Y. J. Cho and H. Y. Zhou, Demiclosed principle and weak con- vergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001), 1245–1260. (Google Scholar)

C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003), 364–374. (Google Scholar)

R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989. (Google Scholar)

K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. (Google Scholar)

J. U. Jeong, Weak and strong convergence of the Noor iteration process for two asymptotically nonexpansive mappings, J. Appl. Math. Computing 23 (2007), 525–536. (Google Scholar)

S. H. Khan and N. Hussain, Convergence theorems for nonself asymptotically nonexpansive mappings, Compt. Math. Appl. 55 (2008), 2544–2553. (Google Scholar)

K. Nammanee, M. A. Noor and S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 314 (2006), 320–334. (Google Scholar)

M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217–229. (Google Scholar)

Z. Opial, Weak convergence of successive approximations for nonexpansive map- pings, Bull. Amer. Math. Soc. 73 (1967), 591–597. (Google Scholar)

H. K. Pathak, Y. J. Cho and S. M. Kang, Strong and weak convergence theorems for nonself asymptotically peturbed nonexpansive mappings, Nonlinear Anal. 70 (2009), 1929–1938. (Google Scholar)

C. I. Podilchuk and R. J. Mammone, Imagine recovery by convex projections using a least squares constraint, J. Opti. Sci. Am. 7 (1990), 517–521. (Google Scholar)

D. R. Sahu, H. K. Xu and J. C. Yao, Asymptotically strict pseudocontracive mappings in the intermediate sense, Nonlinear Anal. 70 (2009), 3502–3511. (Google Scholar)

Y. Song and R. Chen, Viscosity approximation methods for nonexpansive nonself-mappings, J. Math. Anal. Appl. 321 (2006), 316–326. (Google Scholar)

S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005), 506–517. (Google Scholar)

W. Takahashi and T. Tamura, Convergence theorems for pair of nonexpansive mappings, J. Convex Anal. 5 (1998), 45–48. (Google Scholar)

W. Takahashi and K. Takahashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), 45–57. (Google Scholar)

K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308. (Google Scholar)

L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl. 323 (2006), 550–557. (Google Scholar)


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