Abstract
Abstract The purpose of this article is to study the equivalence between some kinds of implicit and explicit iterative approximations. As applications, we utilize our results to study the approximation problems arising in nonexpansive semigroup, variational inclusions and equilibrium problem.
Highlights
Definition 1.1 Let (X, d) be a metric space and let S: X ® X be a mapping
Theorem 1.6 (Moudafi [10]) Let H be a real Hilbert space, C be a nonempty closed convex subset of H, T: C ® C be a nonexpansive mapping with F(T) ≠ Ø and f: C ®
The purpose of this article is to study the equivalence between some kinds of implicit and explicit iterative approximations approximations
Summary
Definition 1.1 Let (X, d) be a metric space and let S: X ® X be a mapping. S is said to be ψ- weakly contractive (weak contraction, for short) [1], if there exists a continuous and strictly increasing function ψ: [0, +∞) ® [0, +∞) with ψ(t) >0, ∀t Î (0, +∞) and ψ(0) = 0 such that d Sx, Sy ≤ d x, y − ψ d x, y , ∀x, y ∈ X. For any given x Î X, the iterative sequence {Tnx} converges strongly to this fixed point This theorem is one of the generalization of Banach contraction Theorem. In 1980, Reich [6] proved the following: Theorem 1.3 (Reich [6]) Let E be a uniformly smooth Banach space, C be a bounded, closed and convex subset of E and let T: C ® C be a nonexpansive mapping. Theorem 1.6 (Moudafi [10]) Let H be a real Hilbert space, C be a nonempty closed convex subset of H, T: C ® C be a nonexpansive mapping with F(T) ≠ Ø and f: C ®. {xn} converge strongly to a fixed point of T in Hilbert space H In the sequel, this theorem is called Moudafi’s viscosity convergence theorem. We utilize our results to study some approximation problems arising in nonexpansive semigroup, variational inclusions and equilibrium problems
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