A Picard-Mann hybrid iterative process

Safeer Hussain Khan

doi:10.1186/1687-1812-2013-69

Abstract

We introduce a new iterative process which can be seen as a hybrid of Picard and Mann iterative processes. We show that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of Berinde (Iterative Approximation of Fixed Points, 2002) for contractions. We support our analytical proof by a numerical example. We prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. Our weak convergence results are proved when the underlying space satisfies Opial’s condition or has Frechet differentiable norm or its dual satisfies the Kadec-Klee property.

Highlights

Introduction and preliminariesLet C be a nonempty convex subset of a normed space E, and let T : C → C be a mapping.Throughout this paper, N denotes the set of all positive integers, I the identity mapping on C and F(T) the set of all fixed points of T.The Picard or successive iterative process [ ] is defined by the sequence {un}: ⎧⎨u = u ∈ C, ⎩un+ = Tun, n ∈ N. ( . )The Mann iterative process [ ] is defined by the sequence {vn}:⎨v = v ∈ C, ⎩vn+ = ( – αn)vn + αnTvn, n ∈ N, where {αn} is in (, )
We introduce a new process which we call ‘PicardMann hybrid iterative process’
The purpose of this paper is to prove that our process ( . ) converges faster than all of Picard, Mann and Ishikawa iterative processes for contractions in the sense of Berinde [ ]

Summary

Introduction and preliminaries

Let C be a nonempty convex subset of a normed space E, and let T : C → C be a mapping. ) converges faster than all of Picard, Mann and Ishikawa iterative processes for contractions in the sense of Berinde [ ]. Lemma [ ] Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space, and let T : C → C be a nonexpansive mapping. Lemma [ ] Let E be a uniformly convex Banach space satisfying Opial’s condition, and let C be a nonempty closed convex subset of E. Definition Suppose that for two fixed-point iterative processes {un} and {vn}, both converging to the same fixed point p, the error estimates un – p ≤ an for all n ∈ N, vn – p ≤ bn for all n ∈ N, are available where {an} and {bn} are two sequences of positive numbers converging to zero.

It is not difficult to show that

Our process