Common Fixed Points of Asymptotically Quasi-Nonexpansive Mappings in CAT(0) Spaces

Abstract

In this manuscript, we investigate and approximate common fixed points of two asymptotically quasi-nonexpansive mappings in CAT0 spaces. Suppose X is a CAT0 space and C is a nonempty closed convex subset of X. Let T1,T2:C→C be two asymptotically quasi-nonexpansive mappings, and F=FT1∩FT2≔x∈C:T1x=T2x=x≠∅. Let αn,βn be sequences in 01. If the sequence {xn} is generated iteratively by xn+1=1−αnxn⊕αnT1nyn,yn=1−βnxn⊕βnT2nxn,n≥1 and x1∈C is the initial element of the sequence (A). We prove that {xn} converges strongly to a common fixed point of T1 and T2 if and only if limn→∞dxnF=0. (B). Suppose αn and βn are sequences in ε1−ε forsome ε∈01. If X is uniformly convex and if either T2 or T1 is compact, then {xn} converges strongly to some common fixed point of T1 and T2. Our results extend and improve the related results in the literature. We also give an example in support of our main results.