×

Weak and strong convergence of Krasnoselski–Mann iteration for hierarchical fixed point problems. (English) Zbl 1154.47055

The authors study the problem of finding hierarchically a fixed point of a nonexpansive mapping \(T\) with respect to a nonexpansive mapping \(P\), namely, to find \(\widetilde{x} \in \text{Fix\,} T\) such that \(\langle \widetilde{x} - P\widetilde{x}, \widetilde{x} - x\rangle \leq 0 \quad \forall x \in \text{Fix\,} T\), using the Krasnoselskij–Mann iterative algorithm: \(x_{n+1}=(1-\alpha _{n})x_{n}+\alpha _{n}(\sigma _{n}Px_{n}+(1-\sigma _{n})Tx_{n})\), \(n \geq 0\).
A. Moudafi [Inverse Probl. 23, No. 4, 1635–1640 (2007; Zbl 1128.47060)] proved a theorem which the above authors have improved upon, even on the weak convergence results. Furthermore, with special conditions on \(P\), the authors prove a strong convergence result of the hierarchical problem in a Hilbert space.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Citations:

Zbl 1128.47060
PDFBibTeX XMLCite
Full Text: DOI