CONVERGENCE OF KRASNOSELSKII'S ITERATION OF RELATIVELY NONEXPANSIVE MAPPING IN STRICTLY CONVEX SPACES
| Main Author: | V.Sankar Raj*1 & S.Gomathi2 |
|---|---|
| Format: | Article Journal |
| Terbitan: |
, 2018
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| Subjects: | |
| Online Access: |
https://zenodo.org/record/1569091 |
Daftar Isi:
- Let us consider two nonempty closed convex subsets A, B of a strictly convex space and a mapping T : A ∪ B → A ∪ B satisfying T (A) ⊆ B and T (B) ⊆ A and ||T x − T y|| ≤ ||x – y||, for all x ∈ A and y ∈ B. First, we provided sufficient conditions for the existence of fixed point pairs (x∗, y∗) in A×B of T for which the distance between x∗ and y∗ is optimum. It is worth mentioning that, we prove the existence of fixed points without invoking proximal normal structure property[4]. Also, we proved the convergence of krasnoselskii’s iteration of a relatively nonexpansive mapping to a fixed point. The main purpose of this article is to provide sufficient conditions to ensure the existence of a pair (x∗, y∗) of points in (A, B) such that T x∗ = x∗, T y∗ = y∗ for which the distance between the fixed points x∗ and y∗ is optimum in some sense. It is worth to mentioning that our existence theorem does not relay on the proximal normal structure property. Also, we proved the strong convergence of Krasnoselskii’s iteration of T to a fixed point which generalizes a result due to Eldred et.al. [ Eldred et.al., Proximal normal structure and relatively nonexpansive mappings, Studia Math. 171(2005),283-293].