A hybrid iteration for asymptotically strictly pseudocontractive mappings

Rajshree Dewangan,Balwant Singh Thakur,Mihai Postolache

doi:10.1186/1029-242x-2014-374

Abstract

Abstract In this paper, we propose a new hybrid iteration for a finite family of asymptotically strictly pseudocontractive mappings. We also prove that such a sequence converges strongly to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings. Results in the paper extend and improve recent results in the literature. MSC:47H09, 47H10.

Highlights

Let H be a real Hilbert space, C be a nonempty closed convex subset of H
The class of strictly pseudocontractive mappings falls into the one between the class of nonexpansive mappings and that of pseudocontractive mappings
We show that the T is not a λ-strictly pseudocontractive mapping

Summary

Introduction

Let H be a real Hilbert space, C be a nonempty closed convex subset of H. Browder and Petryshyn [ ] showed that if a λ-strict pseudocontractive mapping T has a fixed point in C, starting with an initial x ∈ C, the sequence {xn} generated by the formula xn+ = αxn + ( – α)Txn, where α is a constant such that λ < α < , converges weakly to a fixed point of T. Marino and Xu [ ] proposed the following modification for strict pseudocontractive mappings in which the sequence {xn} is given by the same formula ). Inchan and Nammanee [ ] modified the shrinking projection method for asymptotically strict pseudocontractive mappings, in which the sequence {xn} is generated by the same formula Let C be a bounded closed convex subset of a Hilbert space H and {Ti}Ni= : C → C be a finite family of asymptotically (λi, kn(i))-strict pseudocontractive mappings with Lipschitz constant L(ni) ≥ , i = , , .

We observe that

Since for z