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On an open question of V. Colao and G. Marino presented in the paper 'Krasnoselskii-Mann method for non-self mappings'.
- Source :
-
SpringerPlus . 8/11/2016, Vol. 5 Issue 1, p1-9. 9p. - Publication Year :
- 2016
-
Abstract
- Let H be a Hilbert space and let C be a closed convex nonempty subset of H and $$T : C\rightarrow H$$ a non-self nonexpansive mapping. A map $$h : C\rightarrow R$$ defined by $$h(x) := \inf \{\lambda \ge 0 : \lambda x+(1-\lambda )Tx \in C \}$$ . Then, for a fixed $$x_0 \in C$$ and for $$\begin{aligned}{\alpha _0} = \max \left\{ {\frac{1}{2},h({x_0})} \right\}\end{aligned}$$ , Krasnoselskii-Mann algorithm is defined by $$x_{n+1}=\alpha _n+(1-\alpha _n)Tx_n,$$ where $$\alpha _{n+1}=\max \{\alpha _n, h(x_{x_{n+1}})\}$$ . Recently, Colao and Marino (Fixed Point Theory Appl 2015:39, 2015) have proved both weak and strong convergence theorems when C is a strictly convex set and T is an inward mapping. Meanwhile, they proposed a open question for a countable family of non-self nonexpansive mappings. In this article, authors will give an answer and will prove the further generalized results with the examples to support them. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 21931801
- Volume :
- 5
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- SpringerPlus
- Publication Type :
- Academic Journal
- Accession number :
- 117380344
- Full Text :
- https://doi.org/10.1186/s40064-016-2977-8