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

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→ET:K→E be a nonexpansive non-self map with F(T):={x∈K:Tx=x}≠∅F(T):={x∈K:Tx=x}≠∅. Suppose {xn}{xn} is generated iteratively by View the MathML sourcex1∈K,xn+1=P((1-αn)xn+αnTP[(1-βn)xn+βnTxn]), Turn MathJax on n⩾1,n⩾1, where {αn}{αn} and {βn}{βn} are real sequences in [ε,1-ε][ε,1-ε] for some ε∈(0,1)ε∈(0,1). (1) If the dual E*E* of E has the Kadec–Klee property, then weak convergence of {xn}{xn} to some x*∈F(T)x*∈F(T) is proved; (2) If T satisfies condition (A)(A), then strong convergence of {xn}{xn} to some x*∈F(T)x*∈F(T) is obtained.