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Some unique fixed point theorems for rational contractions in partially ordered metric spaces

Muhammad Arshad1*, Erdal Karapınar2 and Jamshaid Ahmad3

Author Affiliations

1 Department of Mathematics, International Islamic University, H-10, Islamabad, 44000, Pakistan

2 Department of Mathematics, Atilim University, İncek, Ankara, 06836, Turkey

3 Department of Mathematics, COMSATS Institute of Information Technology, Chack Shahzad, Islamabad, 44000, Pakistan

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Journal of Inequalities and Applications 2013, 2013:248  doi:10.1186/1029-242X-2013-248


The electronic version of this article is the complete one and can be found online at: http://www.journalofinequalitiesandapplications.com/content/2013/1/248


Received:6 February 2013
Accepted:2 May 2013
Published:17 May 2013

© 2013 Arshad et al.; licensee Springer

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

Abstract

In this paper, we prove some unique fixed point results for an operator T satisfying certain rational contraction condition in a partially ordered metric space. Our results generalize the main result of Jaggi (Indian J. Pure Appl. Math. 8(2):223-230, 1977). We give several examples to show that our results are proper generalization of the existing one.

MSC: 47H10, 54H25, 46J10, 46J15.

Keywords:
fixed point; rational contractions; partially ordered metric spaces

1 Introduction

Fixed point theory is one of the famous and traditional theories in mathematics and has a broad set of applications. In this theory, contraction is one of the main tools to prove the existence and uniqueness of a fixed point. Banach’s contraction principle, which gives an answer on the existence and uniqueness of a solution of an operator equation <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M1">View MathML</a>, is the most widely used fixed point theorem in all of analysis. This principle is constructive in nature and is one of the most useful tools in the study of nonlinear equations. There are many generalizations of Banach’s contraction mapping principle in the literature [1-6]. These generalizations were made either by using the contractive condition or by imposing some additional conditions on an ambient space. There have been a number of generalizations of metric spaces such as rectangular metric spaces, pseudo metric spaces, fuzzy metric spaces, quasi metric spaces, quasi semi-metric spaces, probabilistic metric spaces, D-metric spaces and cone metric spaces

The basic topological properties of ordered sets were discussed by Wolk [7] and Monjardet [8]. The existence of fixed points in partially ordered metric spaces was considered by Ran and Reurings [9]. After this paper, Nieto et al.[10-12] published some new results. Recently, many papers have been reported on partially ordered metric spaces (see, e.g., [9-19] and also [8,20-33]).

The triple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a> is called partially ordered metric spaces (POMS) if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M3">View MathML</a> is a partially ordered set and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4">View MathML</a> is a metric space. Further, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4">View MathML</a> is a complete metric space, the triple <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M6">View MathML</a> is called partially ordered complete metric spaces (POCMS). Throughout the manuscript, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M7">View MathML</a>. A partially ordered metric space <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a> is called ordered complete (OC) if for each convergent sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M9">View MathML</a>, the following condition holds: either

• if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is a non-increasing sequence in X such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M11">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M12">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M13">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M14">View MathML</a>, or

• if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is a non-decreasing sequence in X such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M11">View MathML</a> implies <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M17">View MathML</a><a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M18">View MathML</a>, that is, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M19">View MathML</a>.

In this manuscript, we prove that an operator T satisfying certain rational contraction condition has a fixed point in a partially ordered metric space. Our results generalize the main result of Jaggi [34].

2 Main results

We start this section with the following definition.

Definition 1 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a> be a partially ordered metric space. A self-mapping T on X is called an almost Jaggi contraction if it satisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M21">View MathML</a>

(1)

for any distinct <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M25">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>.

Theorem 2Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space. Suppose that a self-mappingTis an almost Jaggi contraction, continuous and non-decreasing. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>. ThenThas a unique fixed point.

Proof Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a> and set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M31">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M33">View MathML</a>, then T has a fixed point. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34">View MathML</a> is a fixed point of T. So, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M35">View MathML</a> for all n. Since <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M37">View MathML</a>

(2)

Now

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M38">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M39">View MathML</a>

By the triangle inequality, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40">View MathML</a> we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M41">View MathML</a>

(3)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M42">View MathML</a>. Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a> in the inequality (3), we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M44">View MathML</a>. Thus, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is Cauchy. Since X is complete, there exists a point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M47">View MathML</a>. Furthermore, the continuity of T in X implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M48">View MathML</a>

Therefore, z is a fixed point of T in X. Now, if there exists another point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M49">View MathML</a> in X such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M50">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M51">View MathML</a>

a contradiction. Hence u is a unique fixed point of T in X. □

Example 3 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52">View MathML</a> with the usual metric and usual order ≤. We define an operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M54">View MathML</a>

Then T is continuous and non-decreasing. Take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M55">View MathML</a>. Then, for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>, we have the result. Let us examine in detail. Without loss of generality, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58">View MathML</a>.

Case 01. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M59">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M60">View MathML</a>

holds for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M61">View MathML</a> and any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M63">View MathML</a>. Thus, all the conditions of Theorem 2 are satisfied.

Case 02. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M64">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M65">View MathML</a>

holds for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a> and any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M56">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M63">View MathML</a>. Hence, all the conditions of Theorem 2 are satisfied.

Case 03. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M69">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M70">View MathML</a>, then we can easily evaluate that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M71">View MathML</a>. Further, we have <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M72">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M73">View MathML</a>. By the help of these observations, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M74">View MathML</a>

Notice that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M75">View MathML</a> is the fixed point of T.

Definition 4 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a> be a partially ordered metric space. A self-mapping T on X is called a Jaggi contraction if it satisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M77">View MathML</a>

(4)

for any distinct <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a>, where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M80">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>.

Corollary 5Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space. Suppose that a self-mappingTis a Jaggi contraction, continuous and non-decreasing. Suppose that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>. ThenThas a fixed point.

Proof Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85">View MathML</a> in Theorem 2. □

Example 6 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M86">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M87">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M88">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4">View MathML</a> is a complete metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M91">View MathML</a>

Also, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a> iff <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93">View MathML</a>. Clearly, T is an increasing and continuous self-mapping on X. We shall prove that conditions of Corollary 5 hold and T has a fixed point.

Proof For the proof of this example, we have the following cases.

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M94">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M95">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M96">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M97">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M98">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M99">View MathML</a>

• Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M100">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M101">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M102">View MathML</a>

that is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M103">View MathML</a>

Then conditions of Corollary 5 hold and T has a fixed point (here, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M104">View MathML</a> is a fixed point of T). □

In the next theorem, we establish the existence of a unique fixed point of a map T by assuming only the continuity of some iteration of T.

Theorem 7Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space. Suppose that a self-mappingTis non-decreasing and an almost Jaggi contraction. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36">View MathML</a>. If the operator<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108">View MathML</a>is continuous for some positive integerp, thenThas a unique fixed point.

Proof As in Theorem 2, we define a sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109">View MathML</a> and conclude that the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109">View MathML</a> converges to some point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46">View MathML</a>. Thus its subsequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M112">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M113">View MathML</a>) also converges to z. Also,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M114">View MathML</a>

Therefore z is a fixed point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108">View MathML</a>. We now show that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116">View MathML</a>. Let m be the smallest positive integer such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M117">View MathML</a> but <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M118">View MathML</a> (<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M119">View MathML</a>). If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M120">View MathML</a>, then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M121">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M122">View MathML</a>

Regarding (1), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M123">View MathML</a>

Inductively, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M124">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M42">View MathML</a>. Notice that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M126">View MathML</a>. Therefore,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M127">View MathML</a>

a contradiction. Hence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116">View MathML</a>. The uniqueness of z follows as in Theorem 2. □

Corollary 8Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space. Suppose that a self-mappingTis non-decreasing and a Jaggi contraction. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36">View MathML</a>. If the operator<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M108">View MathML</a>is continuous for some positive integerp, thenThas a unique fixed point.

Proof Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85">View MathML</a> in Theorem 7. □

The following theorem generalizes Theorem 2.

Theorem 9Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that for some positive integerm, self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M135">View MathML</a>

(5)

for any distinct<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a>and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M138">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a>. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M142">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143">View MathML</a>is continuous, thenThas a unique fixed point.

Proof Due to Theorem 2, we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143">View MathML</a> has a unique fixed point, say <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46">View MathML</a>. Consider now

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M146">View MathML</a>

Thus, Tz is also a fixed point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143">View MathML</a>. But, by Theorem 2, we know that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143">View MathML</a> has a unique fixed point z. It follows that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M149">View MathML</a>. Hence, z is the unique fixed point of T. □

Corollary 10Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that for some positive integerm, the self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M151">View MathML</a>

(6)

for all distinct<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152">View MathML</a>and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M153">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M142">View MathML</a>. If<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M143">View MathML</a>is continuous, thenThas a unique fixed point.

Proof Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85">View MathML</a> in Theorem 9. □

Now, we give the following example.

Example 11 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52">View MathML</a> with the usual metric and usual order ≤. We define an operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M160">View MathML</a> as follows:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M161">View MathML</a>

It can be easily seen that T is discontinuous and does not satisfy (1) for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M162">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M163">View MathML</a> when <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M164">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M165">View MathML</a>. Now <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M166">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M167">View MathML</a>. It can be verified that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M168">View MathML</a> satisfies the conditions of Theorem 9 and 0 is a unique fixed point of <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M168">View MathML</a>.

Theorem 12Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that a self-mappingTonXsatisfies the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M171">View MathML</a>

(7)

for any points<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173">View MathML</a>, and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M174">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a>. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M30">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M36">View MathML</a>. ThenThas a fixed point.

Proof Define sequences <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M109">View MathML</a> as in Theorem 2. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32">View MathML</a> for some <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M33">View MathML</a> then T has a fixed point. In particular, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34">View MathML</a> is a fixed point of T. Therefore, we assume that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M183">View MathML</a>

(8)

Due to (7), we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M184">View MathML</a>

which implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M185">View MathML</a>

Recursively, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M186">View MathML</a>

As in Theorem 2, we prove that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is a Cauchy sequence. Indeed, by the triangle inequality, we have for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M189">View MathML</a>

(9)

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M190">View MathML</a>. Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a>, then the right-hand side of the inequality (9) tends to 0. Thus, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is Cauchy.

Since X is complete, there exists a <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M194">View MathML</a>

(10)

Consider (7)

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M195">View MathML</a>

(11)

Letting <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a> in (11), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M197">View MathML</a>

which is possible only if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M198">View MathML</a>. Thus, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116">View MathML</a>.

Now, we show that z is the unique fixed point of T. Assume, on the contrary, that the operator T has another fixed point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M200">View MathML</a>. Keeping (7) in mind, we obtain that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M201">View MathML</a>

a contradiction. Hence z is a unique fixed point of T in X. □

Corollary 13Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that a self-mappingTonXsatisfies the condition

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M203">View MathML</a>

(12)

for any points<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173">View MathML</a>, and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M174">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>. ThenThas a fixed point.

Proof Set <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M85">View MathML</a> in Theorem 12. □

Example 14 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M87">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M211">View MathML</a>

Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M4">View MathML</a> is a complete metric space. Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53">View MathML</a> be defined by

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M214">View MathML</a>

Also, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a> iff <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M217">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M218">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>. Clearly, T is an injective, continuous and sequentially convergent mapping on X. We shall prove that conditions of Corollary 8 hold and T has a fixed point.

Proof For the proof of this example, we have the following cases.

Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M220">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M221">View MathML</a>

That is,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M222">View MathML</a>

Hence, conditions of Corollary 8 hold and T has a fixed point (here <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M104">View MathML</a> is a fixed point of T). □

3 Further results

Theorem 15Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing, continuous self-mapping defined onX. Suppose that a self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M225">View MathML</a>

(13)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M228">View MathML</a>andλ, μare non-negative reals such that<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229">View MathML</a>. If there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>, thenThas a fixed point.

Proof By assumption, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a> with <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M234">View MathML</a>, then the proof is finished. So, we suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M235">View MathML</a>. Since T is a non-decreasing mapping, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M236">View MathML</a>

(14)

by iteration. Put <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M31">View MathML</a>. If there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M238">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M32">View MathML</a>, then from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M240">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M34">View MathML</a> is a fixed point, and the proof is finished. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M35">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243">View MathML</a>. Since the points <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M244">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M245">View MathML</a> are comparable for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243">View MathML</a> due to (14), we have the following two cases.

Case 1. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M247">View MathML</a>, then using the contractive condition (13), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M248">View MathML</a>

Hence, we derive that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M249">View MathML</a>

where <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M250">View MathML</a>. Moreover, by the triangular inequality, we have, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M40">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M252">View MathML</a>

and this proves that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M253">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M254">View MathML</a>.

So, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M10">View MathML</a> is a Cauchy sequence and, since X is a complete metric space, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M257">View MathML</a>. Further, the continuity of T implies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M258">View MathML</a>

Thus z is a fixed point.

Case 2. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M259">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M260">View MathML</a>. This implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M261">View MathML</a>, a contradiction. Thus there exists a fixed point z of T. □

Example 16 Let <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M52">View MathML</a> with the usual metric and usual order ≤. We define an operator <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M53">View MathML</a> in the following way:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M264">View MathML</a>

(15)

It is clear that T is continuous on <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M265">View MathML</a>. Now, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M266">View MathML</a> and any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M267">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229">View MathML</a>. Without loss of generality, we assume that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M93">View MathML</a>. So, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M270">View MathML</a>

for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152">View MathML</a>. Also, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M272">View MathML</a> such that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M273">View MathML</a>

is satisfied. This shows that conditions of Theorem 15 hold and T has a fixed point <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M274">View MathML</a>.

We may remove the continuity criteria on T in Theorem 15 as follows.

Theorem 17Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that a self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M276">View MathML</a>

(16)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M279">View MathML</a>andλ, μare non-negative reals with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229">View MathML</a>. And also suppose thatXhas the (OC) property. If there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>, thenThas a fixed point.

Proof We only have to check that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M149">View MathML</a>. As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M284">View MathML</a> is a non-decreasing sequence and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M285">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M286">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243">View MathML</a>. Since T is a non-decreasing mapping, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M288">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M289">View MathML</a> or, equivalently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M290">View MathML</a> for all <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M243">View MathML</a>. Moreover, as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M292">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M293">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M294">View MathML</a>. Suppose that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M295">View MathML</a>. Using a similar argument as that in the proof of Theorem 15 for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>, we obtain that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M297">View MathML</a> is a non-decreasing sequence and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M298">View MathML</a> for certain <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M299">View MathML</a>. Again, using (OC), we have that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M300">View MathML</a>. Moreover, from <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M301">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M302">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M304">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303">View MathML</a> because <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M306">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303">View MathML</a> as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M244">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M309">View MathML</a> are comparable and distinct for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303">View MathML</a>.

Case 1. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M311">View MathML</a>, then applying the contractive condition (16), we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M312">View MathML</a>

Making <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a> in the above inequality, we obtain

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M314">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M315">View MathML</a>, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M316">View MathML</a>, thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M317">View MathML</a>. Particularly, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M318">View MathML</a> and consequently, <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M319">View MathML</a> which is a contradiction. Hence, we conclude that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116">View MathML</a>.

Case 2. If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M321">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M322">View MathML</a>. Taking the limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a>, we get <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M324">View MathML</a>. Then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M318">View MathML</a>, which implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M319">View MathML</a>, a contradiction. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M116">View MathML</a>. □

Now we prove the sufficient condition for the uniqueness of the fixed point in Theorem 15 and Theorem 17, that is,

U: for any <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M328">View MathML</a>, there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M329">View MathML</a> which is comparable to y and z.

Theorem 18Adding the above mentioned condition to the hypothesis of Theorem 15 (or Theorem 17), one obtains the uniqueness of the fixed point ofT.

Proof We distinguish two cases.

Case 1. If y and z are comparable and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M330">View MathML</a>. Now we have two subcases that are as follows:

(i) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M331">View MathML</a>, then using the contractive condition, we have

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M332">View MathML</a>

As <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M315">View MathML</a>, so by the last inequality, we have a contradiction. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334">View MathML</a>.

(ii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M335">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M336">View MathML</a>, a contradiction. Thus <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334">View MathML</a>.

Case 2. If y and z are not comparable, then by a given condition there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M329">View MathML</a> comparable to y and z. Monotonicity implies that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M339">View MathML</a> is comparable to <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M340">View MathML</a> and <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M341">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M342">View MathML</a> .

If there exists <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M343">View MathML</a> such that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M344">View MathML</a>, then as y is a fixed point, the sequence <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M345">View MathML</a> is constant, and consequently <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M346">View MathML</a>. On the other hand, if <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M347">View MathML</a> for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M303">View MathML</a>. Now we have two subcases as follows:

(i) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M349">View MathML</a>, then using the contractive condition, we obtain, for <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M350">View MathML</a>,

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M351">View MathML</a>

This implies that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M352">View MathML</a>

By induction we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M353">View MathML</a>

Taking limit as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M43">View MathML</a> in the above inequality, we get

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M355">View MathML</a>

as <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M229">View MathML</a>. Using a similar argument, we can prove that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M357">View MathML</a>

Now, the uniqueness of the limit gives that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334">View MathML</a>.

(ii) If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M359">View MathML</a>, then <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M360">View MathML</a>. Then

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M361">View MathML</a>

Using a similar argument, we can prove that

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M362">View MathML</a>

Now, the uniqueness of the limit gives that <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M334">View MathML</a>. This completes the proof. □

Remark 19 If in Theorem 15-Theorem 18 <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M364">View MathML</a>, then we obtain Theorem 2.1-Theorem 2.3 of [10].

We get the following fixed point theorem in partially ordered metric spaces if we take <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M365">View MathML</a> in the theorems of Section 3.

Theorem 20Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that a self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M367">View MathML</a>

(17)

for all<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M152">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M58">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M228">View MathML</a>andμis a non-negative real with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M371">View MathML</a>. Suppose also that eitherTis continuous orXsatisfies the condition (OC). If there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>, thenThas a fixed point.

If <a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M3">View MathML</a> satisfies the condition used in Theorem 18, then the uniqueness of a fixed point can be proved.

4 Applications

In this section we state some applications of the main results. The first result is the consequence of Theorem 2.

Corollary 21Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be aT-orbitally complete partially ordered metric space and letTbe a non-decreasing self-mapping defined onX. Suppose that a self-mappingTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M376">View MathML</a>

(18)

for all distinct<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a>and for<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M379">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a>. If there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>thenThas at least one fixed point.

Similarly, the following result is the consequence of Corollary 5.

Corollary 22LetTbe a continuous, non-decreasing self-map defined on a complete partially ordered metric space<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>. Suppose thatTsatisfies the following condition:

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M385">View MathML</a>

(19)

for any distinct<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M22">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M23">View MathML</a>, where<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M80">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>. Suppose there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>. ThenThas a fixed point.

The following result is the consequence of Theorem 12.

Corollary 23Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M2">View MathML</a>be a partially ordered metric space. Let<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M393">View MathML</a>be a non-decreasing, continuous mapping. Suppose that a self-mappingTsatisfies

<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M394">View MathML</a>

(20)

for any<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M172">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M173">View MathML</a>, and for some<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M397">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M26">View MathML</a>and<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M24">View MathML</a>. Suppose that there exists<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M28">View MathML</a>with<a onClick="popup('http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.journalofinequalitiesandapplications.com/content/2013/1/248/mathml/M29">View MathML</a>. ThenThas a fixed point.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors express their gratitude to the anonymous referees for constructive and useful remarks, comments and suggestions.

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