Some convergence theorems of the Mann iteration for monotone α-nonexpansive mappings

In this paper, we introduce the concept of monotone α-nonexpansive mappings in an ordered Banach space E with the partial order ≤, which contains monotone nonexpansive mappings as special case. With the help of the Mann iteration, we show some existence theorems of fixed points of monotone α-nonexpansive mappings in uniformly convex ordered Banach space. Also, we prove some weak and strong convergence theorems of the Mann iteration for finding an order fixed point of monotone α-nonexpansive mappings under the condition lim supn→∞βn(1−βn)>0orlim infn→∞βn(1−βn)>0.