Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense

Volume 9, Issue 7, pp 5119--5135 http://dx.doi.org/10.22436/jnsa.009.07.14

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Authors

Gurucharan S. Saluja - Department of Mathematics, Govt. N. P. G. College of Science, Raipur-492010 (C. G.), India. Mihai Postolache - China Medical University, No.91, Hsueh-Shih Road, Taichung, Taiwan. - Department of Mathematics & Informatics, University "Politehnica" of Bucharest, Splaiul Independentei 313, Bucharest 060042, Romania. Adrian Ghiura - Department of Mathematics & Informatics, University , Splaiul Independentei 313, Bucharest 060042, Romania.

Abstract

We provide a new two-step iteration scheme of mixed type for two asymptotically nonexpansive self mappings in the intermediate sense and two asymptotically nonexpansive non-self mappings in the intermediate sense and establish some strong and weak convergence theorems for mentioned scheme and mappings in uniformly convex Banach spaces. Our results extend and generalize the corresponding results of Chidume et al. [C. E. Chidume, E. U. Ofoedu, H. Zegeye, J. Math. Anal. Appl., 280 (2003), 364-374] and [C. E. Chidume, N. Shahzad, H. Zegeye, Numer. Funct. Anal. Optim., 25 (2004), 239-257], Guo et al. [W. Guo, W. Guo, Appl. Math. Lett., 24 (2011), 2181-2185] and [W. Guo, Y. J. Cho, W. Guo, Fixed Point Theory Appl., 2012 (2012), 15 pages], Saluja [G. S. Saluja, J. Indian Math. Soc. (N.S.), 81 (2014), 369-385], Schu [J. Schu, Bull. Austral. Math. Soc., 43 (1991), 153-159], Tan and Xu [K. K. Tan, H. K. Xu, J. Math. Anal. Appl., 178 (1993), 301-308], Wang [L. Wang, J. Math. Anal. Appl., 323 (2006), 550-557], Wei and Guo [S. I. Wei, W. Guo, Commun. Math. Res., 31 (2015), 149-160] and [S. Wei, W. Guo, J. Math. Study, 48 (2015), 256-264].

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Keywords

• Asymptotically nonexpansive self and non-self mapping in intermediate sense
• new two-step iteration scheme of mixed type
• common fixed point
• uniformly convex Banach space
• strong convergence
• weak convergence.

MSC

•  47J25
•  47H09
•  65J15

References

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• [3] C. E. Chidume, N. Shahzad, H. Zegeye, Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense, Numer. Funct. Anal. Optim., 25 (2004), 239--257


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• [15] S. Wei, W. Guo, Weak convergence theorems for mixed type asymptotically nonexpansive mappings, J. Math. Study, 48 (2015), 256--264