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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 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.
 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.
 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.
 K. Goebel W.A. Kirk. Topics in Metric Fixed Point Theory Cambridge Studies in Advanced Mathematics vol. 28 Cambridge University Press Cambridge 1990.
 B. Halpern Fixed points of nonexpanding maps Bull. Amer. Math. Soc. 73 (1967) 957-961.
 S. He C. Yang P. Duan Realization of the hybrid method for Mann iterations Appl. Math. Comp. 217 (2010) 4239-4247.
 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.
 P.L. Lions Approximation de points fixes de contractions C. R. Acad. Sci. Ser. A-B Paris 284 (1977) 1357-1359.
 P.E. Maingé The viscosity approximation process for quasi-nonexpansive mappings in Hilbert spaces Comput. Math. Appl. 59 (2010) 74-79.
 A. Moudafi Viscosity approximations methods for fixed point problems J. Math. Anal. Appl. 241 (2000) 46-55.
 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.
 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.
 H.K. Xu Viscosity approximations methods for nonexpansive mappings J. Math. Anal. Appl. 298 (2004) 279-291.
 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.