Strong and weak convergence of Mann iteration of monotone α -nonexpansive mappings in uniformly convex Banach spaces

In this paper, the demiclosed principle of monotone α -nonexpansive mapping is showed in a uniformly convex Banach space with the partial order ≤ . With the help of such a demiclosed principle, the strong convergence of Mann iteration of monotone α -nonexpansive mapping T are proved without some compact conditions such as semi-compactness of T , and the weakly convergent conclusions of such an iteration are studied without the conditions such as Opial s condition. These convergent theorems are obtained under the iterative coefficient satisfying the condition, +∞∑k=1min{αk,(1−αk)}=+∞, which contains α k = 1/ k + 1 as a special case.