On the strong convergence theorems by the hybrid method for a family of mappings in uniformly convex Banach spaces

Subject Areas : Applied Mathematics

M. Salehi ¹ (Department of Mathematics, Islamic Azad University, Savadkooh Branch, Savadkooh, Iran.)
V. Dadashi ² (Department of Mathematics, Islamic Azad University, Sari Branch, Sari, Iran.)
M. Roohi ³ (Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, Babolsar, Iran.)

Received: 2015-10-05 Accepted : 2015-10-05 Published : 2012-12-01

Keywords:

Abstract :

Some algorithms for nding common xed point of a family of mappings isconstructed. Indeed, let C be a nonempty closed convex subset of a uniformlyconvex Banach space X whose norm is Gateaux dierentiable and let {Tn} bea family of self-mappings on C such that the set of all common fixed pointsof {Tn} is nonempty. We construct a sequence {xn} generated by the hybridmethod and also we give the conditions of {Tn} under which {xn} convergesstrongly to a common xed point of {Tn}.

References:

[1] S. Atsushiba and W. Takahashi, Strong convergence theorems for nonexpansive
semigroups by a hybrid method, J. Nonlinear Convex Anal. 3 (2002), 231{242.
[2] H. H. Bauschke and P. L. Combettes, A weak-to-strong convergence principle for
fejer-monotone methods in Hilbert spaces, Math. Oper. Res. 26 (2001), 248{264.
[3] K. Goebel and S. Reich, Uniform convexity, hyperbolic geometry, and nonexpan-
sive mappings, Marcel Dekker, New York, 1984.
[4] Y. Haugazeau, Sur les inequations variationnelles et la minimisation de fonc-
tionnelles convexes, Universite de Paris, Paris, France, 1968.
[5] H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of varia-
tional inequalities for monotone mappings, Panamer. Math. J. 14 (2004), 49{61.
[6] K. Nakajo, K. Shimoji and W. Takahashi, Strong convergence theorems by the
hybrid method for families of mappings in Banach spaces, Nonlinear Anal.(TMA)
71 (2009), 812{818.
[7] K. Nakajo, K. Shimoji and W. Takahashi, Strong convergence theorems by the
hybrid method for families of nonexpansive mappings in Hilbert spaces, Taiwanese
J. Math. 10 (2006), 339{360.
[8] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive
mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372{
379.
[9] M. V. Solodov and B. F. Svaiter, Forcing strong convergence of proximal point
iterations in a Hilbert space, Math. Program. Ser. A 87 (2000), 189{202.
[10] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama,
2000.
[11] H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal.
(TMA) 16 (1991), 1127{1138.