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

Zheng, Yuchun¹ ; Wang, Lin ²

¹ College of Statistics and Mathematics, Yunnan University of Finance and Economics, Longquan Road, Kunming, 650221, P. R. China;School of Mathematics and Information Science, Henan Normal University, XinXiang HeNan, 453007, P. R. China
² College of Statistics and Mathematics, Yunnan University of Finance and Economics, Longquan Road, Kunming, 650221, P. R. China

Journal of nonlinear sciences and its applications, Tome 11 (2018) no. 9, p. 1085-1095.

Résumé

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

Détail

DOI : 10.22436/jnsa.011.09.07

Classification : 47H06, 47J05, 47J25, 47H10, 47H17, 49J40, 65J15
Keywords: Ordered Banach space, fixed point, monotone \(\alpha\)-nonexpansive mapping, strong convergence

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