Convergece Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings

Let E be a real Banach space, K a closed convex nonempty subset of E, and T1,T2, ￠ € |,Tm:K ￠ ’K asymptotically quasi-nonexpansive mappings with sequences (resp.) {kin}n=1 ￠ satisfying kin ￠ ’1 as n ￠ ’ ￠ , and ￠ ‘n=1 ￠ (kin ￠ ’1)< ￠ , i=1,2, ￠ € |,m. Let { ±n}n=1 ￠ be a sequence in [ μ, ￠ € ‰1 ￠ ’ μ], ￠ € ‰ ￠ € ‰ μ ￠ (0,1). Define a sequence {xn} by x1 ￠ K, xn+1=(1 ￠ ’ ±n)xn+ ±nT1nyn+m ￠ ’2, yn+m ￠ ’2=(1 ￠ ’ ±n)xn+ ±nT2nyn+m-3, ￠ € |, yn=(1 ￠ ’ ±n)xn+ ±nTmnxn, n ￠ ‰ ￥1, ￠ € ‰ ￠ € ‰m ￠ ‰ ￥2. Let ￠ i=1mF(Ti) ￠ ‰ ￠ …. Necessary and sufficient conditions for a strong convergence of the sequence {xn} to a common fixed point of the family {Ti}i=1m are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.