Iterative Approximation of Common Fixed Points of Two Nonself Asymptotically Nonexpansive Mappings

Abstract EN

Suppose that K is nonempty closed convex subset of a uniformly convex and smooth Banach space E with P as a sunny nonexpansive retraction and F:=F(T1)∩F(T2)={x∈K:T1x=T2x=x}≠∅.

Let T1,T2:K→E be two weakly inward nonself asymptotically nonexpansive mappings with respect to P with two sequences {kn(i)}⊂[1,∞) satisfying ∑n=1∞(kn(i)-1)<∞(i=1,2), respectively.

For any given x1∈K, suppose that {xn} is a sequence generated iteratively by xn+1=(1-αn)(PT1)nyn+αn(PT2)nyn, yn=(1-βn)xn+βn(PT1)nxn, n∈N, where {αn} and {βn} are sequences in [a,1-a] for some a∈(0,1).

Under some suitable conditions, the strong and weak convergence theorems of {xn} to a common fixed point of T1 and T2 are obtained.