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 {αn}n=1 be a sequence in [ , 1− ], ∈ (0,1). Define a sequence {xn} by x1 ∈ K , xn+1 = (1− αn)xn + αnT n 1 yn+m−2, yn+m−2 = (1− αn)xn + αnT 2 yn+m−3, . . . , yn = (1− αn)xn + αnT mxn, n ≥ 1, m≥ 2. Let ⋂i=1F(Ti) =∅. Necessary and sufficient conditions for a strong convergence of the sequence {xn} to a common fixed point of the family {Ti}i=1 are proved. Under some appropriate conditions, strong and weak convergence theorems are also proved.