Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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CONVERGENCE OF MODIFIED MULTI-STEP ITERATIVE FOR A FINITE FAMILY OF ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS

  • Xiao, Juan (School of Mathematics and Statistics Southwest University) ;
  • Deng, Lei (School of Mathematics and Statistics Southwest University) ;
  • Yang, Ming-Ge (College of Mathematics Science Luoyang Normal University)
  • Received : 2013.06.09
  • Published : 2014.01.31
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Abstract

In a uniformly convex Banach space, we introduce a iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings and utilize a new inequality to prove several convergence results for the iterative sequence. The results generalize and unify many important known results of relevant scholars.

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References

  1. S. S. Chang, H. W. Joseph Lee, and C. K. Chan, On Reich's strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Anal. 66 (2007), no. 11, 2364-2374. https://doi.org/10.1016/j.na.2006.03.025
  2. C. E. Chidume, Iterative algorithms for nonexpansive mappings and some of their generalizations, Nonlinear analysis and applications: to V. Lakshmikantham on his 80th birthday. Vol. 1, 2, 383-429, Kluwer Acad. Publ., Dordrecht, 2003.
  3. C. E. Chidume and B. Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330 (2007), no. 1, 377-387. https://doi.org/10.1016/j.jmaa.2006.07.060
  4. C. E. Chidume, J. L. Li, and A. Udomene, Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 133 (2005), no. 2, 473-480. https://doi.org/10.1090/S0002-9939-04-07538-0
  5. C. E. Chidume, E. U. Ofoedu, and H. Zegeye, Strong and weak convergence theorem for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 280 (2003), no. 2, 364-374. https://doi.org/10.1016/S0022-247X(03)00061-1
  6. C. E. Chidume and N. Shahzad, Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings, Nonlinear Anal. 62 (2005), no. 6, 1149-1156. https://doi.org/10.1016/j.na.2005.05.002
  7. Y. J. Cho, H. Y. Zhou, and G. Guo, Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47 (2004), no. 4-5, 707-717. https://doi.org/10.1016/S0898-1221(04)90058-2
  8. K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  9. A. R. Khan and M. A. Ahmed, Convergence of a general iterative scheme for a finite family of asymptotically quasi-nonexpansive mappings in convex metric spaces and applications, J. Appl. Math. Comput. 59 (2010), no. 8, 2990-2995. https://doi.org/10.1016/j.camwa.2010.02.017
  10. A. R. Khan, A. A. Domlo, and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 1-11. https://doi.org/10.1016/j.jmaa.2007.06.051
  11. H. Kiziltunc and S. Temir, Convergence theorems by a new iteration process for a finite family of nonself asymptotically nonexpansive mappings with errors in Banach spaces, J. Appl. Math. Comput. 61 (2011), no. 9, 2480-2489. https://doi.org/10.1016/j.camwa.2011.02.030
  12. M. O. Osilike and S. C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000), no. 10, 1181-1191. https://doi.org/10.1016/S0895-7177(00)00199-0
  13. B. E. Rhoades and S. M. Soltuz, The equivalence between Mann-Ishikawa iterations and multistep iteration, Nonlinear Anal. 58 (2004), no. 1-2, 219-228. https://doi.org/10.1016/j.na.2003.11.013
  14. J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), no. 1, 153-159. https://doi.org/10.1017/S0004972700028884
  15. S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 311 (2005), no. 2, 506-517. https://doi.org/10.1016/j.jmaa.2005.03.002
  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), no. 1, 351-358. https://doi.org/10.1016/S0022-247X(03)00537-7
  17. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138. https://doi.org/10.1016/0362-546X(91)90200-K
  18. H. K. Xu and R. Ori, An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), 767-773. https://doi.org/10.1081/NFA-100105317
  19. I. Yildlrlm and M. Ozdemir, A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings, Nonlinear Analy. 71 (2009), no. 3-4, 991-999. https://doi.org/10.1016/j.na.2008.11.017
  20. Y. Zhou and S. S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. Appl. Optim. 23 (2002), no. 7-8, 911-921. https://doi.org/10.1081/NFA-120016276