Abstract
In this paper, we study viscosity approximations with $(\psi,\varphi)$ -weakly contractive mappings. We show that Moudafi’s viscosity approximations follow from Browder and Halpern type convergence theorems. Our results generalize a number of convergence theorems including a strong convergence theorem of Song and Liu (Fixed Point Theory Appl. 2009:824374, 2009).
Highlights
Introduction and preliminariesLet (M, d) be a metric space and f : M → M a self-mapping
A mapping f : M → M is a contraction if there exists r ∈ [, ) such that for all x, y ∈ M, d f (x), f (y) ≤ rd(x, y)
In, Moudafi [ ] generalized Browder’s and Halpern’s theorems and proved that in a real Hilbert space H, for a given u ∈ Y ⊆ H, the sequence {un} generated by the algorithm un+ = αnf + ( – αn)S(un) for n ∈ N ∪ { }, where f : Y → Y is a contraction, S : Y → Y a nonexpansive mapping and {αn} ⊆ (, ), satisfying certain conditions, converges strongly to a fixed point of S in Y, which is the unique solution to the following variational inequality: (I – f )x∗, x∗ – x ≥, ∀x ∈ F(S)
Summary
Introduction and preliminariesLet (M, d) be a metric space and f : M → M a self-mapping. ) Every (ψ, φ)-weakly contractive mapping of a complete metric space has a unique fixed point.
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