A General Viscosity Iterative Method for a Countable Family of Nonexpansive Mappings in Hilbert Spaces
Jintana Joomwong, Somyot Plubtieng · 2008
In this paper, let H be a real Hilbert space and we introduce an iterative sequence as follows for finding a common element of fixed points for a countable family of nonexpansive mappings in a Hilbert space: xn+1 = αnγf(xn) + βnxn + ((1 − βn)I − αnA)Tnxn, n ≥ 0, where γ ≥ 0, f: H − → H is a given contraction mapping. A is a strongly positive bounded linear operater and {Tn} is a sequence of nonexpansive mappings on H. Then, we prove that such a sequence converges strongly to a common fixed point of nonexpansive mappings. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of nonexpansive mappings, the equilibrium problems. This result extends and improves the corresponding result of Marino and Xu, [A general iterative method for nonexpansive mapping