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The convergence theorems of Ishikawa iterative process with errors for Φ-hemi-contractive mappings in uniformly smooth Banach spaces

Zhiqun Xue1*, Guiwen Lv1 and BE Rhoades2

Author Affiliations

1 Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, 050043, P.R. China

2 Department of Mathematics, Indiana University, Bloomington, IN, 47405-7106, USA

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Fixed Point Theory and Applications 2012, 2012:206  doi:10.1186/1687-1812-2012-206

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2012/1/206


Received:11 May 2012
Accepted:29 October 2012
Published:22 November 2012

© 2012 Xue et al.; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let D be a nonempty closed convex subset of an arbitrary uniformly smooth real Banach space E, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> be a generalized Lipschitz Φ-hemi-contractive mapping with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a> be four real sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a> and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M8">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14">View MathML</a> be any bounded sequences in D, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a> be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the fixed point q of T. A related result deals with the operator equations for a generalized Lipschitz and Φ-quasi-accretive mapping.

MSC: 47H10.

Keywords:
generalized Lipschitz mapping; Φ-hemi-contractive mapping; Ishikawa iterative sequence with errors; uniformly smooth real Banach space

1 Introduction and preliminary

Let E be a real Banach space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M16">View MathML</a> be its dual space. The normalized duality mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M17">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M18">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M19">View MathML</a> denotes the generalized duality pairing. It is well known that

(i) If E is a smooth Banach space, then the mapping J is single-valued;

(ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M20">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M22">View MathML</a>;

(iii) If E is a uniformly smooth Banach space, then the mapping J is uniformly continuous on any bounded subset of E. We denote the single-valued normalized duality mapping by j.

Definition 1.1 ([1])

Let D be a nonempty closed convex subset of E, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> be a mapping.

(1) T is called strongly pseudocontractive if there is a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M24">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M26">View MathML</a>

(2) T is called ϕ-strongly pseudocontractive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M28">View MathML</a> and a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M31">View MathML</a>

(3) T is called Φ-pseudocontractive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M25">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M28">View MathML</a> and a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M36">View MathML</a>

It is obvious that Φ-pseudocontractive mappings not only include ϕ-strongly pseudocontractive mappings, but also strongly pseudocontractive mappings.

Definition 1.2 ([1])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> be a mapping and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M38">View MathML</a>.

(1) T is called ϕ-strongly-hemi-pseudocontractive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M39">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M41">View MathML</a> and a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M44">View MathML</a>

(2) T is called Φ-hemi-pseudocontractive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M39">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M41">View MathML</a> and the strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M50">View MathML</a>

Closely related to the class of pseudocontractive-type mappings are those of accretive type.

Definition 1.3 ([1])

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M51">View MathML</a>. The mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52">View MathML</a> is called strongly quasi-accretive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55">View MathML</a> and a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M57">View MathML</a>; T is called ϕ-strongly quasi-accretive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55">View MathML</a> and a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M29">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M30">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M63">View MathML</a>; T is called Φ-quasi-accretive if for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M54">View MathML</a>, there exist <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M55">View MathML</a> and a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M69">View MathML</a>.

Definition 1.4 ([2])

For arbitrary given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, the Ishikawa iterative process with errors <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M71">View MathML</a> is defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M72">View MathML</a>

(1.1)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14">View MathML</a> are any bounded sequences in D; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a> are four real sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a> and satisfy <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M80">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M81">View MathML</a>, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M83">View MathML</a>, then the sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M85">View MathML</a>

(1.2)

is called the Mann iterative process with errors.

Definition 1.5 ([3,4])

A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> is called generalized Lipschitz if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M87">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M88">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M89">View MathML</a>.

The aim of this paper is to prove the convergent results of the above Ishikawa and Mann iterations with errors for generalized Lipschitz Φ-hemi-contractive mappings in uniformly smooth real Banach spaces. For this, we need the following lemmas.

Lemma 1.6 ([5])

LetEbe a uniformly smooth real Banach space, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M90">View MathML</a>be a normalized duality mapping. Then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M91">View MathML</a>

(1.3)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M92">View MathML</a>.

Lemma 1.7 ([6])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M93">View MathML</a>be a nonnegative sequence which satisfies the following inequality:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M94">View MathML</a>

(1.4)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M95">View MathML</a>with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M96">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M97">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M98">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>.

2 Main results

Theorem 2.1LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M100">View MathML</a>be a generalized Lipschitz Φ-hemi-contractive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a>be four real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a>and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M107">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14">View MathML</a>be any bounded sequences inD, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>be an Ishikawa iterative sequence with errors defined by (1.1). Then (1.1) converges strongly to the unique fixed pointqofT.

Proof Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M1">View MathML</a> is a generalized Lipschitz Φ-hemi-contractive mapping, there exists a strictly increasing continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M34">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M35">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M118">View MathML</a>

i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M119">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M120">View MathML</a>

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M121">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M40">View MathML</a>.

Step 1. There exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M124">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M125">View MathML</a> (range of Φ). Indeed, if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M126">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M127">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M128">View MathML</a>; if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M129">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M130">View MathML</a>, then for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M131">View MathML</a>, there exists a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M132">View MathML</a> in D such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M133">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M135">View MathML</a>. Furthermore, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M136">View MathML</a> is bounded. Hence, there exists a natural number <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M137">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M138">View MathML</a> for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M139">View MathML</a>, then we redefine <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M140">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M141">View MathML</a>.

Step 2. For any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a> is bounded. Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M144">View MathML</a>, then from Definition 1.2(2), we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M145">View MathML</a>. Denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M146">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M147">View MathML</a>. Since T is generalized Lipschitz, so T is bounded. We may define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M148">View MathML</a>. Next, we want to prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M149">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M150">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M151">View MathML</a>. Now, assume that it holds for some n, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M149">View MathML</a>. We prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M153">View MathML</a>. Suppose it is not the case, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M154">View MathML</a>. Since J is uniformly continuous on a bounded subset of E, then for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M155">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M156">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M157">View MathML</a> when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M158">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M159">View MathML</a>. Now, denote

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M160">View MathML</a>

Owing to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M161">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>, without loss of generality, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M163">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M82">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>, denote <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M166">View MathML</a>. So, we have

(2.1)

(2.2)

(2.3)

(2.4)

and

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M176">View MathML</a>

Using Lemma 1.6 and the above formulas, we obtain

(2.10)

and

(2.11)

Substitute (2.11) into (2.10)

(2.12)

this is a contradiction. Thus, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M153">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M181">View MathML</a> is a bounded sequence. So, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M182">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M183">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M184">View MathML</a> are all bounded sequences.

Step 3. We want to prove <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>. Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M187">View MathML</a>.

By (2.10), (2.11), we have

(2.13)

and

(2.14)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M190">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M191">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M192">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>.

Taking (2.14) into (2.13),

(2.15)

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M195">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>.

Set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M197">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M198">View MathML</a>. If it is not the case, we assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M199">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M200">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M201">View MathML</a>, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M202">View MathML</a>. Thus, from (2.15) it follows that

(2.16)

This implies that

(2.17)

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M205">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M206">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M207">View MathML</a>. Then we get that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M208">View MathML</a>

Applying Lemma 1.7, we get that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M98">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>. This is a contradiction and so <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M198">View MathML</a>. Therefore, there exists an infinite subsequence such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M212">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M214">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M215">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213">View MathML</a>. In view of the strictly increasing continuity of Φ, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M217">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M219">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M213">View MathML</a>. Next, we want to prove <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M223">View MathML</a>, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M224">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M225">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M226">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M227">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M228">View MathML</a>, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M229">View MathML</a>. First, we want to prove <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M230">View MathML</a>. Suppose it is not the case, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M231">View MathML</a>. Using (1.1), we may get the following estimates:

(2.18)

(2.19)

Since Φ is strictly increasing, then (2.19) leads to <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M234">View MathML</a>. From (2.15), we have

(2.20)

which is a contradiction. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M230">View MathML</a>. Suppose that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M237">View MathML</a> holds. Repeating the above course, we can easily prove that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M238">View MathML</a> holds. Therefore, for any m, we obtain that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M237">View MathML</a>, which means <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M185">View MathML</a> as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>. This completes the proof. □

Theorem 2.2LetEbe an arbitrary uniformly smooth real Banach space, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52">View MathML</a>be a generalized Lipschitz Φ-quasi-accretive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M243">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a>be four real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a>and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M107">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M14">View MathML</a>be any bounded sequences inE, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>be an Ishikawa iterative sequence with errors defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M257">View MathML</a>

(2.21)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258">View MathML</a>is defined by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M259">View MathML</a>for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>converges strongly to the unique solution of the equation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M262">View MathML</a> (or the unique fixed point ofS).

Proof Since T is a generalized Lipschitz and Φ-quasi-accretive mapping, it follows that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M263">View MathML</a>

i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M264">View MathML</a>

i.e.,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M265">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M92">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M267">View MathML</a>. The rest of the proof is the same as that of Theorem 2.1. □

Corollary 2.3LetEbe an arbitrary uniformly smooth real Banach space, Dbe a nonempty closed convex subset ofE, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M100">View MathML</a>be a generalized Lipschitz Φ-hemi-contractive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M2">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>be two real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a>and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M273">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>be any bounded sequence inD, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>be the Mann iterative sequence with errors defined by (1.2). Then (1.2) converges strongly to the unique fixed pointqofT.

Corollary 2.4LetEbe an arbitrary uniformly smooth real Banach space, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M52">View MathML</a>be a generalized Lipschitz Φ-quasi-accretive mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M243">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M6">View MathML</a>be two real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a>and satisfy the conditions (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M273">View MathML</a>as<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M9">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M10">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M11">View MathML</a>. For some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M12">View MathML</a>, let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>be any bounded sequence inE, and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>be the Mann iterative sequence with errors defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M292">View MathML</a>

(2.22)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258">View MathML</a>is defined by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M259">View MathML</a>for any<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M21">View MathML</a>. Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>converges strongly to the unique solution of the equation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M262">View MathML</a> (or the unique fixed point ofS).

Remark 2.5 It is mentioned that in 2006, Chidume and Chidume [1] proved the approximative theorem for zeros of generalized Lipschitz generalized Φ-quasi-accretive operators. This result provided significant improvements of some recent important results. Their result is as follows.

Theorem CC ([[1], Theorem 3.1])

LetEbe a uniformly smooth real Banach space and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M298">View MathML</a>be a mapping with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M299">View MathML</a>. SupposeAis a generalized Lipschitz Φ-quasi-accretive mapping. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M3">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M4">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M5">View MathML</a>be real sequences in<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M7">View MathML</a>satisfying the following conditions: (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M304">View MathML</a>; (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M305">View MathML</a>; (iii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M306">View MathML</a>; (iv) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M307">View MathML</a>. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>be generated iteratively from arbitrary<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M309">View MathML</a>by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M310">View MathML</a>

(2.23)

where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M258">View MathML</a>is defined by<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M312">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M313">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M13">View MathML</a>is an arbitrary bounded sequence inE. Then, there exists<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M315">View MathML</a>such that if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M316">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M317">View MathML</a>, the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M15">View MathML</a>converges strongly to the unique solution of the equation<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M319">View MathML</a>.

However, there exists a gap in the proof process of above Theorem CC. Here, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M320">View MathML</a> (<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M321">View MathML</a>) does not hold in line 14 of Claim 2 on page 248, i.e., <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M322">View MathML</a> is a wrong case. For instance, set the iteration parameters: <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M323">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M324">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M325">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M326">View MathML</a>; <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M327">View MathML</a> . Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M328">View MathML</a>, but <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2012/1/206/mathml/M329">View MathML</a>. Therefore, the proof of above Theorem CC is not reasonable. Up to now, we do not know the validity of Theorem CC. This will be an open question left for the readers!

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in writing this paper. All authors read and approved the final manuscript.

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