Convergence and weaker control conditions for hybrid iterative algorithms

Abstract

Abstract Very 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

Keywords

• Strong convergence
• Variational inequality
• Fixed points
• <it>k</it>-Lipschitzian
• <it>&#951;</it>-strongly monotone
• Hilbert space