Abstract
AbstractVery recently, Yao et al. (Appl. Math. Comput. 216, 822-829, 2010) have proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this article, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. It is worth pointing out that we use a new method to prove our results. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Furthermore, we also give a weak convergence theorem for their iterative algorithm involving an infinite family of nonexpansive mappings in a Hilbert space.MSC: 47H05, 47H09, 47H10
Highlights
Let H be a real Hilbert space and C be a nonempty, closed, convex subset of H, let F : H ® H be a nonlinear operator
The variational inequality problem is formulated as finding a point x* Î C such that
Γ, 1 2 for some γ > 0, they proved that the sequences {xn} and {yn} defined by (1.2) converge strongly to x∗ ∈ ∩∞ n=1F(Tn), which solves the following variational inequality: Fx∗, x − x∗ ≥ 0, ∀x ∈ ∩∞ n=1F(Tn)
Summary
Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. Under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. It is worth pointing out that we use a new method to prove our results. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. We give a weak convergence theorem for their iterative algorithm involving an infinite family of nonexpansive mappings in a Hilbert space.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
