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Some coincidence and common fixed point theorems for ordered Prešić-Reich type contractions

Satish Shukla1, Slobodan Radojević2*, Zorica A Veljković3 and Stojan Radenović2

Author Affiliations

1 Department of Applied Mathematics, Shri Vaishnav Institute of Technology and Science, Gram Baroli, Sanwer Road, Indore, M.P., 453331, India

2 Department of Mathematics, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11120, Serbia

3 Department of Industrial Engineering, Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, Belgrade, 11120, Serbia

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Journal of Inequalities and Applications 2013, 2013:520  doi:10.1186/1029-242X-2013-520

The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/520


Received:17 June 2013
Accepted:24 September 2013
Published:9 November 2013

© 2013 Shukla 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

The purpose of this paper is to prove some coincidence and common fixed point theorems for ordered Prešić-Reich type contractions in ordered metric spaces. Results of this paper generalize and extend several known results from metric spaces into product spaces when the underlying space is an ordered metric space. An example illustrates the case when new results can be applied while old ones cannot.

Keywords:
Prešić type mapping; coincidence point; common fixed point; ordered space

1 Introduction and preliminaries

The well-known Banach contraction mapping principle states that if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M1">View MathML</a> is a complete metric space and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2">View MathML</a> is a self-mapping such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M3">View MathML</a>

(1)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M5">View MathML</a>, then there exists a unique <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M7">View MathML</a>. This point x is called the fixed point of the mapping f. On the other hand, for mappings <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2">View MathML</a>, Kannan [1] introduced the contractive condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M9">View MathML</a>

(2)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M11">View MathML</a> is a constant and proved a fixed point theorem using (2) instead of (1). Conditions (1) and (2) are independent, as it was shown by two examples in [2].

Reich [3], for mappings <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M2">View MathML</a>, generalized Banach and Kannan fixed point theorems using the contractive condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M13">View MathML</a>

(3)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a>, where α, β, γ are nonnegative constants with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M15">View MathML</a>. An example in [3] shows that condition (3) is a proper generalization of (1) and (2).

In 1965, Prešić [4,5] extended the Banach contraction mapping principle to mappings defined on product spaces and proved the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16">View MathML</a>be a complete metric space, kbe a positive integer and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a>be a mapping satisfying the following contractive type condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M18">View MathML</a>

(4)

for every<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M20">View MathML</a>are nonnegative constants such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M21">View MathML</a>. Then there exists a unique point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M23">View MathML</a>. Moreover, if<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M24">View MathML</a>are arbitrary points inXand for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M26">View MathML</a>, then the sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27">View MathML</a>is convergent and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M28">View MathML</a>.

Note that condition (4) in the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M29">View MathML</a> reduces to the well-known Banach contraction mapping principle. So, Theorem 1.1 is a generalization of the Banach fixed point theorem. Some generalizations and applications of the Prešić theorem can be seen in [4-19].

The existence of a fixed point in partially ordered sets was investigated by Ran and Reurings [20] and then by Nieto and Lopez [21,22]. Fixed point results in ordered metric spaces were obtained by several authors (see, e.g., [6,18,23-32]). The following version of the fixed point theorem was proved, among others, in these papers.

Theorem 1.2 (see [22] and references therein)

Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M30">View MathML</a>be a partially ordered set, and letdbe a metric onXsuch that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16">View MathML</a>is a complete metric space. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M32">View MathML</a>be a nondecreasing map with respect to ⪯. Suppose that the following conditions hold:

(i) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M33">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M34">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M36">View MathML</a>;

(ii) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M38">View MathML</a>;

(iii) fis continuous.

Thenfhas a fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M39">View MathML</a>.

Pǎcurar [10] introduced the Prešić-Kannan type contraction and proved some common fixed point theorems for such contractions. Very recently, in [18] (see also [33]) authors introduced the ordered Prešić type contraction and generalized the result of Prešić and proved some fixed point theorems for such mappings. In this paper, we introduce the ordered Prešić-Reich type contraction and prove some common fixed point theorems for such type of mappings in ordered metric spaces. Our results generalize and extend the results of Prešić [4,5], Pǎcurar [10], Malhotra et al.[18], Luong and Thuan [33], Nieto and López [21] and several known results of metric spaces. An example, which illustrates the case when new results can be applied while old ones cannot, is included.

The following definitions will be needed in the sequel.

Definition 1.3 Let X be a nonempty set, k be a positive integer and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a> be a mapping. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M41">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6">View MathML</a> is called a fixed point of f.

Definition 1.4 (see [13])

Let X be a nonempty set, k be a positive integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M43">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M44">View MathML</a> be mappings.

(a) An element <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M6">View MathML</a> is said to be a coincidence point of f and g if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M46">View MathML</a>.

(b) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M47">View MathML</a>, then w is called a point of coincidence of f and g.

(c) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M48">View MathML</a>, then x is called a common fixed point of f and g.

(d) Mappings f and g are said to be commuting if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M49">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M50">View MathML</a>.

(e) Mappings f and g are said to be weakly compatible if they commute at their coincidence points.

Remark that the above definition in the case <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51">View MathML</a> reduces to the usual definitions of commuting and weakly compatible mappings in the sense of [34] (for details, see the Introduction from [34]).

Definition 1.5 (see [18])

Let a nonempty set X be equipped with a partial order ‘⪯’ such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M16">View MathML</a> is a metric space, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53">View MathML</a> is called an ordered metric space. A sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27">View MathML</a> in X is said to be nondecreasing with respect to ‘⪯’ if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M55">View MathML</a> . Let k be a positive integer and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a> be a mapping, then f is said to be nondecreasing with respect to ‘⪯’ if for any finite nondecreasing sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M57">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M58">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59">View MathML</a> be a mapping. f is said to be g-nondecreasing with respect to ‘⪯’ if for any finite nondecreasing sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M60">View MathML</a> we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M61">View MathML</a>.

Remark 1.6 For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51">View MathML</a>, the above definitions reduce to usual definitions of fixed point and nondecreasing mapping in a metric space.

Definition 1.7 Let X be a nonempty set equipped with partial order ‘⪯,’ and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59">View MathML</a> be a mapping. A nonempty subset of X is said to be well ordered if every two elements of are comparable. Elements <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M66">View MathML</a> are called g-comparable if ga and gb are comparable. is called g-well ordered if for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M68">View MathML</a>, a and b are g-comparable, i.e., ga and gb are comparable.

Example 1.8 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M69">View MathML</a>, ‘⪯’ be a partial order relation on X defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M70">View MathML</a>. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M71">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M59">View MathML</a> be defined by <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M73">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M74">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M75">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M76">View MathML</a>. Then it is clear that is not well ordered but it is g-well ordered.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M78">View MathML</a> be an ordered metric space. Let k be a positive integer and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M43">View MathML</a> be a mapping. f is said to be an ordered Prešić type contraction if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M80">View MathML</a>

(5)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M82">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83">View MathML</a> are nonnegative constants such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M84">View MathML</a>. If (5) is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85">View MathML</a>, then f is called a Prešić type contraction.

f is said to be an ordered Prešić-Kannan type contraction (see [10] for details) if f satisfies following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M86">View MathML</a>

(6)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M88">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M89">View MathML</a>. If (6) is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M85">View MathML</a>, then f is called a Prešić-Kannan type contraction.

f is said to be an ordered Prešić-Reich type contraction (see also [16]) if f satisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M91">View MathML</a>

(7)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M82">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95">View MathML</a> are nonnegative constants such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M96">View MathML</a>. If (7) is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a>, then f is called a Prešić-Reich type contraction.

Note that the Prešić-Reich type contraction is a generalization of Prešić type and Prešić-Kannan type contractions. Indeed, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M98">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M99">View MathML</a>, a Prešić-Reich type contraction reduces into a Prešić type contraction and for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M100">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M101">View MathML</a>, and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M102">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M103">View MathML</a>, a Prešić-Reich type contraction reduces into a Prešić-Kannan type contraction. Also, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51">View MathML</a>, a Prešić-Reich type contraction reduces into a Reich contraction, so it generalizes the Banach and Kannan contractions.

Now we can state our main results.

2 Main results

Theorem 2.1Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53">View MathML</a>be an ordered complete metric space. Letkbe a positive integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107">View MathML</a>be two mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M108">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109">View MathML</a>is a closed subset ofXand

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M110">View MathML</a>

(8)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95">View MathML</a>are nonnegative constants such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M115">View MathML</a>

(9)

Suppose that the following conditions hold:

(I) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117">View MathML</a>;

(II) fisg-nondecreasing;

(III) if a nondecreasing sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118">View MathML</a>converges to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122">View MathML</a>.

Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M123">View MathML</a>. Moreover, the set of common fixed points offandgisg-well ordered if and only iffandghave a unique common fixed point.

Proof Starting with given <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>, we define a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M125">View MathML</a> as follows: let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M126">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M127">View MathML</a>. As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M128">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M129">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M130">View MathML</a>. Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M131">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M132">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M133">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M134">View MathML</a>. Again, as f is g-nondecreasing and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M133">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M136">View MathML</a>. Choose <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M137">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M138">View MathML</a> (which is possible since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M139">View MathML</a>). So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M140">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M141">View MathML</a>. Continuing this process, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M142">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M143">View MathML</a>

and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M144">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M145">View MathML</a> . Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M146">View MathML</a> is nondecreasing with respect to ‘⪯,’ that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27">View MathML</a> is g-nondecreasing with respect to ‘⪯.’ We shall show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M148">View MathML</a> is a Cauchy sequence in <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M150">View MathML</a> for any n, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M151">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M152">View MathML</a>, using (8), the above inequality implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M153">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M154">View MathML</a>

since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M150">View MathML</a>. In view of (9), we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M156">View MathML</a>, therefore it follows from the above inequality that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M157">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M158">View MathML</a>. Similarly, it can be shown that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M159">View MathML</a>

Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M125">View MathML</a> is a Cauchy sequence. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M161">View MathML</a> for all n, then for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M162">View MathML</a>, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M163">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27">View MathML</a> is g-nondecreasing, using (8), the above inequality implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M165">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M166">View MathML</a>

Setting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M167">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M168">View MathML</a>

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M169">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M170">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M171">View MathML</a>, then in view of (9) we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M172">View MathML</a>. Therefore

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M173">View MathML</a>

By repeating this process, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M174">View MathML</a>

(10)

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M175">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M176">View MathML</a>, then it follows from (10) that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M177">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M178">View MathML</a>, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M179">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M180">View MathML</a>. Therefore, it follows from the above inequality that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M181">View MathML</a>. Therefore <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M148">View MathML</a> is a Cauchy sequence. As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109">View MathML</a> is closed, there exist <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M184">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M185">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M186">View MathML</a>

We shall show that v is a point of coincidence of f and g. For any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M187">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M188">View MathML</a>

By (III) we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>, also, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M191">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M192">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M193">View MathML</a>. Therefore, using (8) in the above inequality, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M194">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M195">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M196">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M197">View MathML</a>, therefore it follows from the above inequality that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M198">View MathML</a>

(11)

Thus, u is a coincidence point and v is a corresponding point of coincidence of f and g. Suppose, f and g are weakly compatible, then by (11) we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M199">View MathML</a>

Again, by (III), <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M200">View MathML</a>; therefore using (8) and a similar process as several times before, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M201">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M202">View MathML</a>, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M203">View MathML</a>

Thus v is a common fixed point of f and g. Suppose that the set of common fixed points is g-well ordered. We shall show that the common fixed point is unique. Assume on the contrary that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M204">View MathML</a> is another common fixed point of f and g, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M205">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M206">View MathML</a>. As v and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M204">View MathML</a> are g-comparable, let for example <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M208">View MathML</a>. From (8), it follows that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M209">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M202">View MathML</a>, we obtain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M211">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M212">View MathML</a>, a contradiction. Therefore the common fixed point is unique. For converse, if a common fixed point of f and g is unique, then the set of common fixed points of f and g is singleton, and thus g-well ordered. □

Remark 2.2 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213">View MathML</a> be an ordered metric space, and let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M214">View MathML</a> be two mappings. Then f is called an ordered g-weak contraction if

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M215">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M217">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M220">View MathML</a> are nonnegative constants such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M221">View MathML</a>. If the above inequality is satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M4">View MathML</a>, then f is called a g-weak contraction (see [35]). For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M51">View MathML</a> in Theorem 2.1, we get a fixed point result for an ordered g-weak contraction in metric spaces.

The following is a fixed point result for ordered Prešić-Reich type mappings in metric spaces and can be obtained by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M224">View MathML</a> (that is, the identity mapping of X) in Theorem 2.1.

Corollary 2.3Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213">View MathML</a>be an ordered complete metric space. Letkbe a positive integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a>be a mapping such that the following conditions hold:

(I) fis an ordered Prešić-Reich type contraction;

(II) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M228">View MathML</a>;

(III) fis nondecreasing (with respect to ‘);

(IV) if a nondecreasing sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M27">View MathML</a>converges to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M185">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M231">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>.

Thenfhas a fixed point. Moreover, the set of fixed points offis well ordered if and only iffhas a unique fixed point.

The following corollary is a generalization of the result of Prešić in an ordered metric space and can be obtained by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M98">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M234">View MathML</a> in Theorem 2.1.

Corollary 2.4Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213">View MathML</a>be an ordered complete metric space. Letkbe a positive integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107">View MathML</a>be two mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M238">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109">View MathML</a>is a closed subset ofXand

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M240">View MathML</a>

(12)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M83">View MathML</a>are nonnegative constants such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M84">View MathML</a>. Suppose that the following conditions hold:

(I) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117">View MathML</a>;

(II) fisg-nondecreasing;

(III) if a nondecreasing sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118">View MathML</a>converges to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122">View MathML</a>.

Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M123">View MathML</a>. Moreover, the set of common fixed points offandgisg-well ordered if and only iffandghave a unique common fixed point.

The following corollary generalizes the result of Pǎcurar [10] in ordered metric spaces and can be obtained by taking <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M100">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M254">View MathML</a> in Theorem 2.1.

Corollary 2.5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M53">View MathML</a>be an ordered complete metric space. Letkbe a positive integer, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M17">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M107">View MathML</a>be two mappings such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M139">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M109">View MathML</a>is a closed subset ofXand

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M260">View MathML</a>

(13)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M19">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M112">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M95">View MathML</a>are nonnegative constants such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M264">View MathML</a>. Suppose that the following conditions hold:

(I) there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M37">View MathML</a>such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M117">View MathML</a>;

(II) fisg-nondecreasing;

(III) if a nondecreasing sequence<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M118">View MathML</a>converges to<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M119">View MathML</a>, then<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M120">View MathML</a>for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M25">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M122">View MathML</a>.

Thenfandghave a point of coincidence. If, in addition, fandgare weakly compatible, thenfandghave a common fixed point<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M272">View MathML</a>. Moreover, the set of common fixed points offandgisg-well ordered if and only iffandghave a unique common fixed point.

The following example illustrates that an ordered Prešić-Reich type contraction may not be an ordered Prešić type or ordered Prešić-Kannan type or Prešić-Reich type contraction; moreover, that the fixed point of an ordered Prešić-Reich type contraction may not be unique (when the set of fixed points of f is not well-ordered).

Example 2.6 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M273">View MathML</a> and order relation ‘⪯’ be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M274">View MathML</a>

and let d be the usual metric on X. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M213">View MathML</a> is an ordered complete metric space. For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276">View MathML</a>, define <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M277">View MathML</a> by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M278">View MathML</a>

Then:

(a) f is not an ordered Prešić type contraction;

(b) f is not an ordered Prešić-Kannan type contraction;

(c) f is not a Prešić-Reich type contraction;

(d) f is an ordered Prešić-Reich type contraction with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M280">View MathML</a>.

Proof (a) For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276">View MathML</a>, (5) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M282">View MathML</a>

(14)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219">View MathML</a> are nonnegative constants such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M287">View MathML</a>. Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M288">View MathML</a>, therefore for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M289">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M290">View MathML</a>, (14) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M291">View MathML</a>

But <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M292">View MathML</a> and therefore the above inequality will never hold. Thus f is not an ordered Prešić type contraction.

(b) For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276">View MathML</a>, (6) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M294">View MathML</a>

(15)

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284">View MathML</a>, where β is a nonnegative constant such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M297">View MathML</a>. Note that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M298">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M299">View MathML</a> and therefore for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M300">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M301">View MathML</a>, (15) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M302">View MathML</a>

But <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M303">View MathML</a>, and therefore the above inequality will never hold. Thus f is not an ordered Prešić-Kannan type contraction.

(c) For <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M276">View MathML</a>, (7) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M305">View MathML</a>

(16)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M218">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M219">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M308">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M309">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M310">View MathML</a> are nonnegative constants such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M311">View MathML</a>. If f is a Prešić-Reich type contraction, then inequality (16) must be satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M283">View MathML</a>. Note that for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M313">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M314">View MathML</a>, (16) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M315">View MathML</a>

But <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M316">View MathML</a>, and therefore the above inequality will never hold. Thus f is not a Prešić-Reich type contraction.

(d) If f is an ordered Prešić-Reich type contraction, then inequality (16) must be satisfied for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M317">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284">View MathML</a>. Indeed, we have to check the validity of (16) only for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M319">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M320">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M321">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M322">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M321">View MathML</a>, then (16) is satisfied trivially. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M324">View MathML</a> or <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M325">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M284">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M327">View MathML</a>, then (16) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M328">View MathML</a>

which is valid for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M330">View MathML</a>. If any one of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M331">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M332">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M333">View MathML</a> is equal to 1, then with a similar process one can verify the same result. If any two of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M331">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M332">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M333">View MathML</a> are equal to 1, for example, let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M337">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M338">View MathML</a>, then (16) becomes

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M339">View MathML</a>

which is valid for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M279">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M330">View MathML</a>. Similarly, in all possible cases, (16) is satisfied for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M342">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M343">View MathML</a>. Thus, f is an ordered Prešić-Reich type contraction. All the conditions of Corollary 2.3 (except the set of fixed points of f is well ordered) are satisfied and the set of fixed points of f is <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M344">View MathML</a>. Note that the set of fixed points of f, that is ℱ, is not well ordered (as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/520/mathml/M345">View MathML</a>) and fixed point f is not unique. □

Competing interests

All authors of the present paper disclose no actual potential conflict of interests including any financial, personal or other relationships with people or organizations.

Authors’ contributions

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

Acknowledgements

This paper is supported by Grant No. 1740024 from the Ministry of Science and Technical Development of the Republic of Serbia.

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