Strong and weak convergence of Mann iteration of monotone α-nonexpansive mappings in uniformly convex Banach spaces
Yuchun Zheng, Lin Wang · The Journal of Nonlinear Sciences and Applications · 2018
In this paper, the demiclosed principle of monotone \(\alpha\)-nonexpansive mapping is showed in a uniformly convex Banach space with the partial order ``\(\leq\)". With the help of such a demiclosed principle, the strong convergence of Mann iteration of monotone \(\alpha\)-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, \[\sum\limits_{k=1}^{+\infty}\min\{\alpha_k,(1-\alpha_k)\}=+\infty,\] which contains \(\alpha_k=\frac1{k+1}\) as a special case