Mann iteration of Cesàro means for asymptotically non-expansive mappings

Let KK be a nonempty closed convex subset of a uniformly convex Banach space EE with a uniformly Gâteaux differentiable norm. Suppose that T:K→KT:K→K is an asymptotically non-expansive mapping and for arbitrary initial value x0∈Kx0∈K, we will introduce the Mann iteration of its Cesàro means: xn+1=αnxn+(1−αn)1n+1∑j=0nTjxn,n≥0, and prove its strong and weak convergence whenever ∑n=0∞bn<+∞ and {αn}{αn} is a real sequence in (0,1)(0,1) satisfying one of the conditions: either (i) limn→∞αn=0limn→∞αn=0 or (ii) ∑n=0∞αn(1−αn)=+∞ or (iii) 0