A viscosity projection method for class T mappings

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Abstract

In this paper, we firstly introduce a viscosity projection method for the class T mappings

xn+1=αnPH(xn, Snxn) f(xn) + (1-αn)Snxn,

where Sn = (1 - w)I + wTn, w ∈ (0; 1); Tn ∈ T and prove strong convergence theorems of the proposed method. It is verified that the viscosity projection method converges locally faster than the viscosity method. Furthermore, we present a viscosity projection method for a quasi-nonexpansive and nonexpansive mappings in Hilbert spaces. A numerical test provided in the paper shows that the viscosity projection method converges faster than the viscosity method.

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  • [1] H.H. Bauschke P.L. Combettes A weak-to-strong convergence princi­ple for Fejer-monotone methods in Hilbert spaces Math. Oper. Res. 26 (2001) 248-264.

  • [2] P.L. Combettes Quasi-Fejerian analysis of some optimization algorithms in: D. Butnariu Y. Censor S. Reich (Eds.) Inherently Parallel Algo­rithms for Feasibility and Optimization Elsevier New York 2001 pp. 115-152.

  • [3] Q.L. Dong S. He F. Su Strong convergence theorems by shrinking pro­jection methods for class T mappings Fixed Point Theory and Appl. Volume 2011 Article ID 681214 7 pages.

  • [4] K. Goebel W.A. Kirk. Topics in Metric Fixed Point Theory Cambridge Studies in Advanced Mathematics vol. 28 Cambridge University Press Cambridge 1990.

  • [5] B. Halpern Fixed points of nonexpanding maps Bull. Amer. Math. Soc. 73 (1967) 957-961.

  • [6] S. He C. Yang P. Duan Realization of the hybrid method for Mann iterations Appl. Math. Comp. 217 (2010) 4239-4247.

  • [7] P. Kumam S. Plubtieng Viscosity approximation methods for monotone mappings and a countable family of nonexpansive mappings Mathemat- ica Slovaca Math. Slovaca 61 (2) (2011) 257-274.

  • [8] P.L. Lions Approximation de points fixes de contractions C. R. Acad. Sci. Ser. A-B Paris 284 (1977) 1357-1359.

  • [9] P.E. Maingé The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces Comput. Math. Appl. 59 (2010) 74-79.

  • [10] A. Moudafi Viscosity approximations methods for fixed point problems J. Math. Anal. Appl. 241 (2000) 46-55.

  • [11] N. Petrot R. Wangkeeree P. Kumam A viscosity approximation method of common solutions for quasi variational inclusion and fixed point prob­lems Fixed Point Theory 12(1) (2011) 165-178.

  • [12] S. Plubtieng P. Kumam Weak convergence theorem for monotone map­pings and a countable family of nonexpansive mappings J. Comput. Appl. Math. 224 (2009) 614-621.

  • [13] H.K. Xu Viscosity approximations methods for nonexpansive mappings J. Math. Anal. Appl. 298 (2004) 279-291.

  • [14] I. Yamada N. Ogura Hybrid steepest descent method for the variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings Numer. Funct. Anal. Optim. 25 (7-8) (2004) 619-655.

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