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Suzuki-type fixed point theorem for fuzzy mappings in ordered metric spaces

Basit Ali1* and M Abbas12

Author Affiliations

1 Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences, Lahore, 54792, Pakistan

2 Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield, Pretoria, South Africa

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Fixed Point Theory and Applications 2013, 2013:9  doi:10.1186/1687-1812-2013-9

The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2013/1/9


Received:8 September 2012
Accepted:18 December 2012
Published:10 January 2013

© 2013 Ali and Abbas; 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, a Suzuki-type fixed fuzzy point result for fuzzy mappings in complete ordered metric spaces is obtained. As an application, we establish the existence of coincidence fuzzy points and common fixed fuzzy points for a hybrid pair of a single-valued self-mapping and a fuzzy mapping. An example is also provided to support the main result presented herein.

MSC: 47H10, 47H04, 47H07.

Keywords:
fixed fuzzy point; fuzzy mapping; fuzzy set; approximate quantity

1 Introduction and preliminaries

Let X be a space of points with generic elements of X denoted by x and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M1">View MathML</a>. A fuzzy subset of X is characterized by a membership function such that each element in X is associated with a real number in the interval I. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2">View MathML</a> be a metric space and a fuzzy set A in X is characterized by a membership function A. Then α-level set of A, denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M3">View MathML</a>, is defined as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M4">View MathML</a>

for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M5">View MathML</a> and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M6">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M8">View MathML</a> denotes the closure of the non-fuzzy set B. A fuzzy set A in X is said to be an approximate quantity if and only if for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M3">View MathML</a> is a compact, convex subset of X and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M11">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12">View MathML</a> be a family of all approximate quantities in X. A fuzzy set A is said to be more accurate than a fuzzy set B denoted by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M13">View MathML</a> (that is, B includes A) if and only if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M14">View MathML</a> for each x in X, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M15">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M16">View MathML</a> denote the membership function of A and B, respectively. It is easy to see that if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M17">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M18">View MathML</a>.

Corresponding to each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20">View MathML</a>, the fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21">View MathML</a> of X is the fuzzy set <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M22">View MathML</a> given by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M23">View MathML</a>

For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M24">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M25">View MathML</a>

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M26">View MathML</a> be a collection of all fuzzy subsets of X and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12">View MathML</a> be a subcollection of all approximate quantities. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M28">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9">View MathML</a>, define

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M31">View MathML</a>

Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M32">View MathML</a> is a nondecreasing function of α and D is a metric on <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M9">View MathML</a>. Define <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M35">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2">View MathML</a> be a metric space and Y be an arbitrary set. A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M37">View MathML</a> is called a fuzzy mapping, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M38">View MathML</a> for each y in Y. Thus, if we characterize a fuzzy set Fy in a metric space X by a membership function Fy, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M39">View MathML</a> is the grade of membership of x in Fy. Therefore, a fuzzy mapping F is a fuzzy subset of <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M40">View MathML</a> with a membership function <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M39">View MathML</a>.

In a more general sense than that given in [1], a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M42">View MathML</a> is a fuzzy mapping over X[2] and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M43">View MathML</a> is the fixed degree of x in <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M44">View MathML</a>.

Definition 1 ([3])

A fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21">View MathML</a> in X is called a fixed fuzzy point of the fuzzy mapping F if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M47">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M48">View MathML</a>. That is, the fixed degree of x in Fx is at least α. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M49">View MathML</a>, then x is a fixed point of a fuzzy mapping F.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M50">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M51">View MathML</a>.

A fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21">View MathML</a> in X is called a coincidence fuzzy point of the hybrid pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M54">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M55">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M56">View MathML</a>. That is, the fixed degree of gx in Fx is at least α. A fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M21">View MathML</a> in X is called a common fixed fuzzy point of the hybrid pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M59">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M60">View MathML</a> (the fixed degree of x and gx in Fx is the same and is at least α).

We denote by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M61">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M62">View MathML</a> the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a>, respectively.

A hybrid pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a> is called w-fuzzy compatible if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M65">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M66">View MathML</a>.

A mapping g is called F-fuzzy weakly commuting at some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20">View MathML</a> if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M68">View MathML</a>.

Lemma 1 ([4])

LetXbe a nonempty set and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M69">View MathML</a>. Then there exists a subset<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M70">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M71">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M72">View MathML</a>is one-to-one.

Definition 2 Let X be a nonempty set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73">View MathML</a> is called an ordered metric space if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2">View MathML</a> is a metric space and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M75">View MathML</a> is partially ordered.

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M75">View MathML</a> be a partially ordered set. Then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a> are said to be comparable if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M78">View MathML</a> or <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M79">View MathML</a> holds.

Define

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M80">View MathML</a>

An ordered metric space is said to satisfy the order sequential limit property if <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M81">View MathML</a> for all n, whenever a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M82">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83">View MathML</a> for all n.

A mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84">View MathML</a> is said to be an ordered fuzzy mapping if the following conditions are satisfied:

(a) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M85">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M86">View MathML</a>.

(b) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M88">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M89">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M90">View MathML</a>.

The following lemmas are needed in the sequel.

Lemma 2 (Heilpern [1])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2">View MathML</a>be a metric space, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M28">View MathML</a>:

1. if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M94">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M95">View MathML</a>;

2. <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M96">View MathML</a>;

3. if<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M95">View MathML</a>, then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M98">View MathML</a>.

Lemma 3 (Lee and Cho [5])

Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M2">View MathML</a>be a complete metric space andFbe a fuzzy mapping fromXinto<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M12">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M101">View MathML</a>. Then there exists an<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M102">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M103">View MathML</a>.

Zadeh [6] introduced the concept of a fuzzy set. Heilpern [1] introduced the concept of fuzzy mappings in a metric space and proved a fixed point theorem for fuzzy contraction mappings as a generalization of the fixed point theorem for multivalued mappings given by Nadler [7]. Estruch and Vidal [3] proved a fixed point theorem for fuzzy contraction mappings in complete metric spaces which in turn generalizes the Heilpern fixed point theorem. Further generalizations of the result given in [3] were proved in [8,9]. Recently, Suzuki [10] generalized the Banach contraction principle and characterized the metric completeness property of an underlying space. Among many generalizations (see [11-13]) of the results given in [10], Dorić and Lazović [14] obtained Suzuki-type fixed point results for a generalized multivalued contraction in complete metric spaces.

On the other hand, the existence of fixed points in ordered metric spaces has been introduced and applied by Ran and Reurings [15]. Fixed point theorems in partially ordered metric spaces are hybrid of two fundamental principles: Banach contraction theorem with a contractive condition for comparable elements and a selection of an initial point to generate a monotone sequence. For results concerning fixed points and common fixed points in partially ordered metrics spaces, we refer to [16-22].

The aim of this paper is to investigate Suzuki-type fixed point results for fuzzy mappings in complete ordered metric spaces. As an application, a coincidence fuzzy point and a common fixed fuzzy point of the hybrid pair of a single-valued self-mapping and a fuzzy mapping are obtained. We provide an example to support the result.

Throughout this paper, let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M104">View MathML</a> be the nonincreasing function defined by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M105">View MathML</a>

(1)

2 Main results

The following theorem is the main result of the paper and is a generalization of [[14], Theorem 2.1] for fuzzy mappings in ordered metric spaces.

Theorem 4Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73">View MathML</a>be a complete ordered metric space. If an ordered fuzzy mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M108">View MathML</a>

(2)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M110">View MathML</a>

Then there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46">View MathML</a>provided thatXsatisfies the order sequential limit property.

Proof Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M113">View MathML</a> be a real number such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M114">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M116">View MathML</a> is nonempty and compact, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M117">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M118">View MathML</a>

By the given assumption, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M119">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M120">View MathML</a> is nonempty and compact, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M121">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M122">View MathML</a>

Also, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M123">View MathML</a>. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M124">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M125">View MathML</a>

That is,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M126">View MathML</a>

So, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M127">View MathML</a>

Note that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M128">View MathML</a>. If not, then the above inequality gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M129">View MathML</a>

a contradiction. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M130">View MathML</a>. Continuing this process, we construct a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M131">View MathML</a> in X such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M132">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M133">View MathML</a> with

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M134">View MathML</a>

By the given assumption, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M136">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M124">View MathML</a>, so

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M138">View MathML</a>

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M139">View MathML</a>

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M140">View MathML</a>. If not, then by the above inequality, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M141">View MathML</a>

a contradiction as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M142">View MathML</a>. So, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M143">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M144">View MathML</a>

(3)

Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M131">View MathML</a> is a Cauchy sequence in X. Since X is complete, there is some point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M146">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M147">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M83">View MathML</a> for all n, then by the assumption, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M81">View MathML</a>. Now, we show that for every pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M150">View MathML</a> with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M152">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M147">View MathML</a>, there exists a positive integer <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M154">View MathML</a> such that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M155">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M156">View MathML</a>

(4)

Now, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M157">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M158">View MathML</a>

implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M159">View MathML</a>

which on taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160">View MathML</a> gives

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M161">View MathML</a>

If

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M162">View MathML</a>

then

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M164">View MathML</a>

(5)

Now, we show that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165">View MathML</a> for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M166">View MathML</a>. First, consider the case <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M167">View MathML</a>. Assume on the contrary that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M168">View MathML</a>, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M169">View MathML</a>. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M170">View MathML</a>, as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M171">View MathML</a> is nonempty and compact, so for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M172">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M173">View MathML</a>

(6)

Now, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M170">View MathML</a> implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M175">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M176">View MathML</a>. From (5) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M177">View MathML</a>

(7)

Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M178">View MathML</a>

implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M179">View MathML</a>

Hence,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M180">View MathML</a>

which further implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M181">View MathML</a>

We claim that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M182">View MathML</a>. If not, then the above inequality becomes

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M183">View MathML</a>

a contradiction, so we deduce that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M184">View MathML</a>. From inequality (7), we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M185">View MathML</a>

Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M186">View MathML</a>

a contradiction. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165">View MathML</a>.

Now, when <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M188">View MathML</a>, we first prove that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M189">View MathML</a>

(8)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M150">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M191">View MathML</a>, then (8) holds trivially. So, assume that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151">View MathML</a>. For every <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M193">View MathML</a>, one may find a sequence <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M194">View MathML</a> such that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M195">View MathML</a>

As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M194">View MathML</a>, this implies <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M197">View MathML</a>. Using (7) we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M198">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M193">View MathML</a>. If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M200">View MathML</a>, then

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M201">View MathML</a>

This implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M202">View MathML</a>

Hence, for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M203">View MathML</a>, we obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M204">View MathML</a>

On taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M206">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M207">View MathML</a>, then

On taking the limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M210">View MathML</a>

By the given assumption, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M211">View MathML</a>

Thus, for any <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M151">View MathML</a>, (8) holds true. Put <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M213">View MathML</a> in the above inequality to obtain

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M214">View MathML</a>

as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M215">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M216">View MathML</a>. Hence by Lemma 2, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M165">View MathML</a>. □

Corollary 5Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73">View MathML</a>be a complete ordered metric space. If an ordered fuzzy mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M220">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M222">View MathML</a>

Then there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46">View MathML</a>provided thatXsatisfies the order sequential limit property.

Corollary 6Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73">View MathML</a>be a complete ordered metric space. If an ordered fuzzy mapping<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M84">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M227">View MathML</a>

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M229">View MathML</a>

and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M230">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M231">View MathML</a>. Then there exists a point<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M20">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M46">View MathML</a>provided thatXsatisfies the order sequential limit property.

3 An application

Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M50">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M51">View MathML</a>. A pair <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a> is said to be an ordered fuzzy hybrid pair if the following conditions are satisfied:

(c) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M237">View MathML</a> implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M86">View MathML</a>.

(d) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a> gives <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M88">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M241">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M242">View MathML</a>.

(e) <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M243">View MathML</a> whenever <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a>.

Theorem 7Let<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M73">View MathML</a>be a complete ordered metric space. If an ordered fuzzy hybrid pair<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M53">View MathML</a>satisfies

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M248">View MathML</a>

(9)

for all<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M250">View MathML</a>

Then<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M251">View MathML</a>provided thatXsatisfies the order sequential limit property and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M252">View MathML</a>for eachα. Moreover, Fandghave a common fixed fuzzy point if any of the following conditions holds:

(f) Fandgarew-fuzzy compatible, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M253">View MathML</a>and<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M254">View MathML</a>for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M66">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256">View MathML</a>andgis continuous atu.

(g) gisF-fuzzy weakly commuting for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M257">View MathML</a>and is a fixed point ofg, that is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M258">View MathML</a>.

(h) gis continuous atxfor some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M257">View MathML</a>and for some<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256">View MathML</a>such that<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M261">View MathML</a>.

Proof By Lemma 1, there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M70">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M72">View MathML</a> is one-to-one and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M71">View MathML</a>. Define a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M265">View MathML</a> by

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M266">View MathML</a>

(10)

As g is one-to-one on E, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267">View MathML</a> is well defined. Also,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M268">View MathML</a>

(11)

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M87">View MathML</a>. Therefore,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M270">View MathML</a>

for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M243">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267">View MathML</a> satisfies (2) and all the conditions of Theorem 4. Using Theorem 4 with a mapping <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267">View MathML</a>, it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267">View MathML</a> has a fixed fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M275">View MathML</a>. Now, it is left to prove that F and g have a coincidence fuzzy point. Since <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M267">View MathML</a> has a fixed fuzzy point <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M277">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M278">View MathML</a>. As <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M252">View MathML</a>, so there exists <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115">View MathML</a> such that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M281">View MathML</a>, thus it follows that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M282">View MathML</a>. This implies that <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M115">View MathML</a> is a coincidence fuzzy point of F and g. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M251">View MathML</a>. Suppose now that (f) holds. Then for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M286">View MathML</a>, where <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256">View MathML</a>. Thus <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M288">View MathML</a>. Since g is continuous at u, we have that u is a fixed point of g. As F and g are w-fuzzy compatible, and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M289">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M290">View MathML</a>. That is, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M291">View MathML</a> for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M290">View MathML</a>. Now,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M293">View MathML</a>

implies that

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M294">View MathML</a>

On taking limit as <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M160">View MathML</a>, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M296">View MathML</a> and therefore <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M297">View MathML</a>. By Lemma 2 we obtain <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M298">View MathML</a>. Consequently, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M299">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M300">View MathML</a> is a common fixed fuzzy point of F and g. Suppose now that (g) holds. If for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285">View MathML</a>, g is F-fuzzy weakly commuting and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M258">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M303">View MathML</a>. Hence, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M304">View MathML</a> is a common fixed fuzzy point of F and g. Suppose now that (h) holds and assume that for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M285">View MathML</a> and for some <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M256">View MathML</a>, <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M261">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M308">View MathML</a>. By the continuity of g at x and y, we get <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M60">View MathML</a>. The result follows. □

Example 1 Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M310">View MathML</a> be endowed with the usual metric. Let <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M311">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M312">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M313">View MathML</a>. Define a fuzzy mapping F from X into <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M314">View MathML</a> as

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M315">View MathML</a>

and for <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M316">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M317">View MathML</a>

Define a self-map <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M69">View MathML</a> by <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M319">View MathML</a>. Then

Note that for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M322">View MathML</a>

Also, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M323">View MathML</a>, we have

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M324">View MathML</a>

And

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M325">View MathML</a>

If <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M326">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M327">View MathML</a>, then <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M328">View MathML</a>. So, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a>, with <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M330">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M331">View MathML</a>. Hence, for all <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M77">View MathML</a>,

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M333">View MathML</a>

hold true, where

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M334">View MathML</a>

and

<a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M335">View MathML</a>

Hence, all the conditions of Theorem 7 are satisfied. Moreover, for each <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M336">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M337">View MathML</a> and <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M338">View MathML</a>. For <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M24">View MathML</a>, we have <a onClick="popup('http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.fixedpointtheoryandapplications.com/content/2013/1/9/mathml/M340">View MathML</a>.

4 Conclusion

The Banach contraction principle has become a classical tool to show the existence of solutions of functional equations in nonlinear analysis (see for details [23-26]). Suzuki-type fixed point theorems [10,14] are the generalizations of the Banach contraction principle that characterize metric completeness of underlying spaces. Fuzzy sets and mappings play important roles in the process of fuzzification of systems. Suzuki-type fixed point theorems for fuzzy mappings obtained in this article can further be used in the process of finding the solutions of functional equations involving fuzzy mappings in fuzzy systems. In the main result, we not only extended the mapping to a fuzzy mapping, but also the underlying metric space has been replaced with ordered metric spaces. In this article, we defined coincidence fuzzy points and common fixed fuzzy points of the hybrid pair of a single-valued self-mapping and a fuzzy mapping and applied our main result to obtain the existence of coincidence fuzzy points and common fixed fuzzy points of the hybrid pair.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the final manuscript.

Acknowledgements

The authors are thankful to the referees for their critical remarks which helped to improve the presentation of this paper.

References

  1. Heilpern, S: Fuzzy mappings and fuzzy fixed point theorems. J. Math. Anal. Appl.. 83, 566–569 (1981). Publisher Full Text OpenURL

  2. Sen, CS: Fixed degree for fuzzy mappings and a generalization of Ky Fan’s theorem. Fuzzy Sets Syst.. 24, 103–112 (1987). Publisher Full Text OpenURL

  3. Estruch, VD, Vidal, A: A note on fixed fuzzy points for fuzzy mappings. Rend. Ist. Mat. Univ. Trieste. 32, 39–45 (2001)

  4. Haghi, RH, Rezapour, S, Shahzad, N: Some fixed point generalizations are not real generalizations. Nonlinear Anal.. 74, 1799–1803 (2011). Publisher Full Text OpenURL

  5. Lee, BS, Cho, SJ: A fixed point theorem for contractive type fuzzy mappings. Fuzzy Sets Syst.. 61, 309–312 (1994). Publisher Full Text OpenURL

  6. Zadeh, LA: Fuzzy sets. Inf. Control. 8, 103–112 (1965)

  7. Nadler, SB Jr..: Multivalued contraction mappings. Pac. J. Math.. 30, 475–488 (1969). Publisher Full Text OpenURL

  8. Sedghi, S, Shobe, N, Altun, I: A fixed fuzzy point for fuzzy mappings in complete metric spaces. Math. Commun.. 13, 289–294 (2008)

  9. Turkoglu, D, Rhoades, BE: A fixed fuzzy point for fuzzy mapping in complete metric spaces. Math. Commun.. 10, 115–121 (2005)

  10. Suzuki, T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc.. 136, 1861–1869 (2008)

  11. Altun, I, Erduran, A: A Suzuki type fixed-point theorem. Int. J. Math. Math. Sci.. 2011, (2011) Article ID 736063. doi:10.1155/2011/736063

  12. Ćirić, L, Abbas, M, Rajović, M, Ali, B: Suzuki type fixed point theorems for generalized multi-valued mappings on a set endowed with two b-metrics. Appl. Math. Comput.. 219, 1712–1723 (2012). Publisher Full Text OpenURL

  13. Singh, SL, Mishra, SN: Coincidence theorems for certain classes of hybrid contractions. Fixed Point Theory Appl.. 2010, (2010) Article ID 898109

  14. Dorić, D, Lazović, R: Some Suzuki type fixed point theorems for generalized multivalued mappings and applications. Fixed Point Theory Appl.. 2011, (2011) Article ID 40

  15. Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some application to matrix equations. Proc. Am. Math. Soc.. 132, 1435–1443 (2004). Publisher Full Text OpenURL

  16. Abbas, M, Khamsi, MA, Khan, AR: Common fixed point and invariant approximation in hyperbolic ordered metric spaces. Fixed Point Theory Appl.. 2011, (2011) Article ID 25. doi:10.1186/1687-1812-2011-25

  17. Amini-Harandi, A, Emami, H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal.. 72, 2238–2242 (2010). Publisher Full Text OpenURL

  18. Ćirić, L, Abbas, M, Saadati, R, Hussain, N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput.. 217, 5784–5789 (2011). Publisher Full Text OpenURL

  19. Harjani, J, Sadarangani, K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal.. 71, 3403–3410 (2009). Publisher Full Text OpenURL

  20. Kadelburg, Z, Pavlović, M, Radenović, S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl.. 59, 3148–3159 (2010). Publisher Full Text OpenURL

  21. Nieto, JJ, Lopez, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22, 223–239 (2005). Publisher Full Text OpenURL

  22. Samet, B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal.. 72, 4508–4517 (2010). Publisher Full Text OpenURL

  23. Baskaran, R, Subrahmanyam, PV: A note on the solution of a class of functional equations. Appl. Anal.. 22, 235–241 (1986). Publisher Full Text OpenURL

  24. Bellman, R: Methods of Nonlinear Analysis. Vol. II, Academic Press, New York (1973)

  25. Bellman, R, Lee, ES: Functional equations in dynamic programming. Aequ. Math.. 17, 1–18 (1978). Publisher Full Text OpenURL

  26. Bhakta, PC, Mitra, S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl.. 98, 348–362 (1984). Publisher Full Text OpenURL