STRONG CONVERGENCE THEOREMS ON THE MODIFIED ITERATIVE ALGORITHM FOR A FAMILY OF FINITE NONEXPANSIVE NONSELF MAPPINGS
Yonghong Yao · 2007
Let K be a nonempty closed convex subset of a real Banach space E which has a uniformly Gâteaux differentiable norm. Assume that K is a sunny nonexpansive retract of E with Q as the sunny nonexpansive retraction. Let Ti : K → E, i = 1, 2, · · · , N be a family of nonexpansive mappings which are weakly inward with F = ⋂N i=1 F (Ti) 6= ∅. Let f : K → K be a fixed contractive mapping. For given x0 ∈ K, let {xn} be generated by the algorithm xn+1 = αnf(xn) + βnxn + γnQTn+1xn, n ≥ 0. Some sufficient and necessary conditions are proved for a common fixed point of a family of nonexpansive mappings provided Ti, i = 1, 2, · · · , N satisfy some mild conditions.