An answer to the Halpern’s open question
Yongfu Su · 2007
Let E be a uniformly smooth Banach space and C be a nonempty closed convex subset of E.Let T:C→C be a nonexpansive mapping such that F(T)≠φ.Given u ∈ C and the initial guess x0 ∈ C is chosen arbitrarily.Let {αn}be a real sequence in(0,1) satisfying the conditions limn→∞αn=0 and ∞∑=∞n=1αn.If he sequence {xn } defined by x n +1=αnu+(1-αn)Txn also satisfies the condition ■Tzn-xn‖-‖zn-xn■=ο(αn),where zn=αnu+(1-αn)Tz n,then {xn}n∞=0 strongly converges to a fixed point of T.