General iteration scheme for finding the common fixed points of an infinite family of nonexpansive mappings
Authors
Guangrong Wu
 School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China.
Liping Yang
 School of Applied Mathematics, Guangdong University of Technology, Guangzhou 510520, China.
Abstract
The purpose of this paper is to suggest and analyze the general viscosity iteration scheme for an infinite
family of nonexpansive mappings \(\{T_i\}^\infty_{i=1}\). Additionally, it proves that this iterative scheme converges
strongly to a common fixed point of \(\{T_i\}^\infty_{i=1}\) in the framework of reflexive and smooth convex Banach space,
which solves some variational inequality. Results proved in this paper improve and generalize recent known
results in the literature.
Keywords
 Nonexpansive mapping
 general iteration scheme
 contraction
 smooth Banach space.
MSC
References

[1]
G. Cai, C. S. Hu, Strong convergence theorems of a general iterative process for a finite family of \(\lambda_i\)strict pseudocontractions in quniformly smooth Banach spaces, Comput. Math. Appl., 59 (2010), 149160

[2]
R. Glowinski, P. Le Tallec, Augmented Lagrangian and operatorsplitting methods in nonlinear mechanics, SIAM Studies in Applied Mathematics, Philadelphia (1989)

[3]
K. Goebel, W. A. Kirk, Topics in Metric Fixed Point Theory, Studies in Advanced Mathematics, Cambridge University Press, Cambridge (1990)

[4]
G. Marino, H.K. Xu, A general iterative method for nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 318 (2006), 4352

[5]
M. A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl., 251 (2000), 217229

[6]
M. J. Shang, X. L. Qin, Y. F. Su, Strong convergence of Ishikawa iterative method for nonexpansive mappings in Hilbert space, J. Math. Inequal., 1 (2007), 195204

[7]
K. Shimoji, W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math., 5 (2001), 387404

[8]
T. Suzuki, Strong convergence of Krasnoselskii and Mann's sequences for oneparameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227239

[9]
H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl., 298 (2004), 279291

[10]
Y. Yao, Y. C. Liou, R. Chen, A general iterative method for an infinite family of nonexpansive mappings, Nonlinear Anal., 69 (2008), 16441654

[11]
Y. H. Yao, Y.C. Liou, J.C. Yao, Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings, Fixed Point Theory and Appl., 2007 (2007), 12 pages