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This article is part of the series Recent Trends on Fixed Point Theory and its Applications.

Open Access Research

Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

Faik Gürsoy1, Vatan Karakaya2* and Billy E Rhoades3

Author Affiliations

1 Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34220, Turkey

2 Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey

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

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Fixed Point Theory and Applications 2013, 2013:76  doi:10.1186/1687-1812-2013-76


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/76


Received:5 December 2012
Accepted:10 March 2013
Published:28 March 2013

© 2013 Gürsoy 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

In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

MSC: 47H10.

Keywords:
new multi-step iteration; S-iteration; data dependence; contractive-like operator

1 Introduction

Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area, viz., [1-12] among others.

Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost importance. Therefore it is natural to see many works related to the convergence of iteration methods such as [13-22].

Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points, etc. One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [23], Rus et al.[24,25], Berinde [26], Espínola and Petruşel [27], Markin [28], Chifu and Petruşel [29], Olantiwo [30,31], Şoltuz [32,33], Şoltuz and Grosan [34], Chugh and Kumar [35] and the references therein.

This paper is organized as follows. In Section 1 we present a brief survey of some known contractive mappings and iterative schemes and collect some preliminaries that will be used in the proofs of our main results. In Section 2 we show that the convergence of a new multi-step iteration, which is a special case of the Jungck multistep-SP iterative process defined in [36], and S-iteration (due to Agarwal et al.) can be used to approximate the fixed points of contractive-like operators. Motivated by the works of Şoltuz [32,33], Şoltuz and Grosan [34], and Chugh and Kumar [35], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators.

As a background of our exposition, we now mention some contractive mappings and iteration schemes.

In [37] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1">View MathML</a>, there exist real numbers a, b, c satisfying <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M2">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M3">View MathML</a> such that, for each pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M4">View MathML</a>, at least one of the following is true:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M5">View MathML</a>

(1.1)

A mapping T satisfying the contractive conditions (z1), (z2) and (z3) in (1.1) is called a Zamfirescu operator. An operator satisfying condition (z2) is called a Kannan operator, while the mapping satisfying condition (z3) is called a Chatterjea operator. As shown in [13], the contractive condition (1.1) leads to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M6">View MathML</a>

(1.2)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M7">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M8">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9">View MathML</a>, and it was shown that this class of operators is wider than the class of Zamfirescu operators. Any mapping satisfying condition (b1) or (b2) is called a quasi-contractive operator.

Extending the above definition, Osilike and Udomene [20] considered operators T for which there exist real numbers <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M10">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M7">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M13">View MathML</a>

(1.3)

Imoru and Olantiwo [38] gave a more general definition: An operator T is called a contractive-like operator if there exists a constant <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9">View MathML</a> and a strictly increasing and continuous function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M15">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M16">View MathML</a>, such that for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M17">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M18">View MathML</a>

(1.4)

A map satisfying (1.4) need not have a fixed point, even if E is complete. For example, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M19">View MathML</a> and define T by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M20">View MathML</a>

WLOG, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M21">View MathML</a>. Then, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M22">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M23">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M24">View MathML</a>, and (1.4) is automatically satisfied.

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M25">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M26">View MathML</a>.

Define φ by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M27">View MathML</a> for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M28">View MathML</a>. Then φ is increasing, continuous, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M16">View MathML</a>. Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M30">View MathML</a> so that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M31">View MathML</a>.

Therefore

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M32">View MathML</a>

for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M33">View MathML</a>, and (1.4) is satisfied for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M25">View MathML</a>. But T has no fixed point.

However, using (1.4) it is obvious that if T has a fixed point, then it is unique.

From now on, we demand that ℕ denotes the set of all nonnegative integers. Let X be a Banach space, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M35">View MathML</a> be a nonempty closed, convex subset of X, and let T be a self-map on E. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M36">View MathML</a> to be the set of fixed points of T. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M37">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M38">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M39">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M40">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M41">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M42">View MathML</a> be real sequences in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M43">View MathML</a> satisfying certain conditions.

In [5] Rhoades and Şoltuz introduced a multi-step iterative procedure given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M44">View MathML</a>

(1.5)

The sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a> defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M46">View MathML</a>

(1.6)

is known as the S-iteration process (see [12,17,39]).

Thianwan [6] defined a two-step iteration <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M47">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M48">View MathML</a>

(1.7)

Recently Phuengrattana and Suantai [7] introduced an SP iteration method defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M49">View MathML</a>

(1.8)

We shall employ the following iterative process. For an arbitrary fixed order <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M42">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M51">View MathML</a>

(1.9)

or, in short,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M52">View MathML</a>

(1.10)

where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M53">View MathML</a>

(1.11)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M54">View MathML</a>

(1.12)

Remark 1 If each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M55">View MathML</a>, then SP iteration (1.8) reduces to two-step iteration (1.7). By taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M57">View MathML</a> in (1.10), we obtain iterations (1.8) and (1.7), respectively.

We shall need the following definition and lemma in the sequel.

Definition 1[40]

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M58">View MathML</a> be two operators. We say that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a> is an approximate operator for T if, for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M60">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M61">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M62">View MathML</a>.

Lemma 1[34]

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M63">View MathML</a>be a nonnegative sequence for which one assumes that there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M64">View MathML</a>such that for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M65">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M66">View MathML</a>

is satisfied, where<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M67">View MathML</a>for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M68">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M69">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M70">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M71">View MathML</a>. Then the following holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M72">View MathML</a>

2 Main results

For simplicity we use the following notation throughout this section.

For any iterative process, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M74">View MathML</a> denote iterative sequences associated to T and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a>, respectively.

Theorem 1Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1">View MathML</a>be a map satisfying (1.4) with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M77">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a>be a sequence defined by (1.10), then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M79">View MathML</a>converges to the unique fixed point of T.

Proof The proof can be easily obtained by using the argument in the proof of ([36], Theorem 3.1). □

This result allows us to give the next theorem.

Theorem 2Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1">View MathML</a>be a map satisfying (1.4) with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M81">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a>be an approximate operator ofTas in Definition 1. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M84">View MathML</a>be two iterative sequences defined by (1.10) with real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M85">View MathML</a>satisfying (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M86">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87">View MathML</a>, (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M88">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M90">View MathML</a>, then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M91">View MathML</a>

Proof For given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M92">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M93">View MathML</a>, we consider the following multi-step iteration for T and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M95">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M96">View MathML</a>

(2.2)

Thus, from (1.4), (2.1) and (2.2), we have the following inequalities.

(2.3)

(2.4)

(2.5)

Combining (2.3), (2.4) and (2.5), we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M100">View MathML</a>

(2.6)

Thus, by induction, we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M101">View MathML</a>

(2.7)

Again, using (1.4), (2.1) and (2.2), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M102">View MathML</a>

(2.8)

Substituting (2.8) in (2.7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M103">View MathML</a>

(2.9)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M105">View MathML</a>, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M107">View MathML</a>

(2.10)

From inequality (2.10) and assumption (i) in (2.9), it follows

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M108">View MathML</a>

(2.11)

Define

From Theorem 1 it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M110">View MathML</a>. Since T satisfies condition (1.4) and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M111">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M112">View MathML</a>

(2.12)

Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M113">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M114">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M87">View MathML</a>, using (1.4) and (1.10), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M116">View MathML</a>

(2.13)

It is easy to see from (2.13) that this result is also valid for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M117">View MathML</a>.

Since φ is continuous, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M118">View MathML</a>

(2.14)

Hence an application of Lemma 1 to (2.11) leads to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M119">View MathML</a>

(2.15)

 □

As shown by Hussain et al. ([22], Theorem 8), in an arbitrary Banach space X, the S-iteration <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a> given by (1.6) converges to the fixed point of T, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1">View MathML</a> is a mapping satisfying condition (1.3).

Theorem 3Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M1">View MathML</a>be a map satisfying (1.4) with<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M81">View MathML</a>, and let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a>be defined by (1.6) with real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M125">View MathML</a>satisfying<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M126">View MathML</a>. Then the sequence<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M45">View MathML</a>converges to the unique fixed point ofT.

Proof The argument is similar to the proof of Theorem 8 of [22], and is thus omitted. □

We now prove the result on data dependence for the S-iterative procedure by utilizing Theorem 3.

Theorem 4LetT, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a>be two operators as in Theorem 2. Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M47">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M74">View MathML</a>be S-iterations defined by (1.6) with real sequences<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M131">View MathML</a>satisfying (i) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M132">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M114">View MathML</a>, and (ii) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M126">View MathML</a>. If<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M89">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M90">View MathML</a>, then we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M137">View MathML</a>

Proof For a given <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M92">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M93">View MathML</a>, we consider the following iteration for T and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M59">View MathML</a>:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M141">View MathML</a>

(2.16)

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M142">View MathML</a>

(2.17)

Using (1.4), (2.16) and (2.17), we obtain the following estimates:

(2.18)

(2.19)

Combining (2.18) and (2.19), we get

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M145">View MathML</a>

(2.20)

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M146">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M9">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M148">View MathML</a>

(2.21)

It follows from assumption (i) that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M149">View MathML</a>

(2.22)

Therefore, combining (2.22) and (2.21) to (2.20) gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M150">View MathML</a>

(2.23)

or, equivalently,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M151">View MathML</a>

(2.24)

Now define

From Theorem 3, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M110">View MathML</a>. Since T satisfies condition (1.4), and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M111">View MathML</a>, using an argument similar to that in the proof of Theorem 2,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M155">View MathML</a>

(2.25)

Using the fact that φ is continuous, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M156">View MathML</a>

(2.26)

An application of Lemma 1 to (2.24) leads to

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M157">View MathML</a>

(2.27)

 □

3 Conclusion

Since the iterative schemes (1.7) and (1.8) are special cases of the iterative process (1.10), Theorem 1 generalizes Theorem 2.1 of [19] and Theorem 2.1 of [18]. By taking <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M57">View MathML</a> in Theorem 2, data dependence results for the iterative schemes (1.8) and (1.7) can be easily obtained. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/76/mathml/M56">View MathML</a>, Theorem 2 reduces to Theorem 3.2 of [35]. Since condition (1.4) is more general than condition (1.3), Theorem 3 generalizes Theorem 8 of [22].

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The first two authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Unit under project number BAPK 2012-07-03-DOP02 for financial support during the preparation of this manuscript.

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