Approximating fixed points of non-self nonexpansive mappings in Banach spaces

Suppose K is a nonempty closed convex nonexpansive retract of a real uniformly convex Banach space E with P as a nonexpansive retraction. Let T:K→E be a nonexpansive non-self map with F(T):={x∈K:Tx=x}≠∅. Suppose {xn} is generated iteratively byx1∈K,xn+1=P((1-αn)xn+αnTP[(1-βn)xn+βnTxn]),n⩾1, where {αn} and {βn} are real sequences in [ε,1-ε] for some ε∈(0,1). (1) If the dual E* of E has the Kadec-Klee property, then weak convergence of {xn} to some x*∈F(T) is proved; (2) If T satisfies condition (A), then strong convergence of {xn} to some x*∈F(T) is obtained.