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

Fixed Point Theory and Applications20132013:9

DOI: 10.1186/1687-1812-2013-9

Received: 8 September 2012

Accepted: 18 December 2012

Published: 10 January 2013

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq1_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq2_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq3_HTML.gif , is defined as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equa_HTML.gif
for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq4_HTML.gif and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq5_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equb_HTML.gif
where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq6_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq3_HTML.gif is a compact, convex subset of X and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equc_HTML.gif

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq8_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq9_HTML.gif (that is, B includes A) if and only if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq10_HTML.gif for each x in X, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq11_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq12_HTML.gif denote the membership function of A and B, respectively. It is easy to see that if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq13_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq14_HTML.gif .

Corresponding to each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq15_HTML.gif , the fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq16_HTML.gif of X is the fuzzy set http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq17_HTML.gif given by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equd_HTML.gif
For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq18_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Eque_HTML.gif
Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq19_HTML.gif be a collection of all fuzzy subsets of X and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq8_HTML.gif be a subcollection of all approximate quantities. For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq7_HTML.gif , define
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equf_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equg_HTML.gif

Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq21_HTML.gif is a nondecreasing function of α and D is a metric on http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq8_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq7_HTML.gif . Define http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq22_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq2_HTML.gif be a metric space and Y be an arbitrary set. A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq23_HTML.gif is called a fuzzy mapping, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq24_HTML.gif for each y in Y. Thus, if we characterize a fuzzy set Fy in a metric space X by a membership function Fy, then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq25_HTML.gif is the grade of membership of x in Fy. Therefore, a fuzzy mapping F is a fuzzy subset of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq26_HTML.gif with a membership function http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq25_HTML.gif .

In a more general sense than that given in [1], a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq27_HTML.gif is a fuzzy mapping over X[2] and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq28_HTML.gif is the fixed degree of x in http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq29_HTML.gif .

Definition 1 ([3])

A fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq16_HTML.gif in X is called a fixed fuzzy point of the fuzzy mapping F if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq30_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq31_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq32_HTML.gif . That is, the fixed degree of x in Fx is at least α. If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq33_HTML.gif , then x is a fixed point of a fuzzy mapping F.

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq35_HTML.gif .

A fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq16_HTML.gif in X is called a coincidence fuzzy point of the hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq37_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq38_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq39_HTML.gif . That is, the fixed degree of gx in Fx is at least α. A fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq16_HTML.gif in X is called a common fixed fuzzy point of the hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq40_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq41_HTML.gif (the fixed degree of x and gx in Fx is the same and is at least α).

We denote by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq43_HTML.gif the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif , respectively.

A hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif is called w-fuzzy compatible if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq44_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq45_HTML.gif .

A mapping g is called F-fuzzy weakly commuting at some point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq15_HTML.gif if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq46_HTML.gif .

Lemma 1 ([4])

Let X be a nonempty set and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq47_HTML.gif . Then there exists a subset http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq48_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq50_HTML.gif is one-to-one.

Definition 2 Let X be a nonempty set. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq51_HTML.gif is called an ordered metric space if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq2_HTML.gif is a metric space and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq52_HTML.gif is partially ordered.

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq52_HTML.gif be a partially ordered set. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif are said to be comparable if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq54_HTML.gif or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq55_HTML.gif holds.

Define
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equh_HTML.gif

An ordered metric space is said to satisfy the order sequential limit property if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq56_HTML.gif for all n, whenever a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq58_HTML.gif for all n.

A mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq59_HTML.gif is said to be an ordered fuzzy mapping if the following conditions are satisfied:
  1. (a)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq60_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq61_HTML.gif .

     
  2. (b)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq63_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq64_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq65_HTML.gif .

     

The following lemmas are needed in the sequel.

Lemma 2 (Heilpern [1])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq2_HTML.gif be a metric space, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq20_HTML.gif :
  1. 1.

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq66_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq67_HTML.gif ;

     
  2. 2.

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq68_HTML.gif ;

     
  3. 3.

    if http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq67_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq69_HTML.gif .

     

Lemma 3 (Lee and Cho [5])

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq2_HTML.gif be a complete metric space and F be a fuzzy mapping from X into http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq70_HTML.gif . Then there exists an http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq71_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq72_HTML.gif .

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 [1113]) 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 [1622].

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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq73_HTML.gif be the nonincreasing function defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ1_HTML.gif
(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 4 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq51_HTML.gif be a complete ordered metric space. If an ordered fuzzy mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq59_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ2_HTML.gif
(2)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equi_HTML.gif

Then there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq15_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq30_HTML.gif provided that X satisfies the order sequential limit property.

Proof Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq74_HTML.gif be a real number such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq75_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq76_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq77_HTML.gif is nonempty and compact, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq78_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equj_HTML.gif
By the given assumption, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq79_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq80_HTML.gif is nonempty and compact, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq81_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equk_HTML.gif
Also, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq82_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq83_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equl_HTML.gif
That is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equm_HTML.gif
So, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equn_HTML.gif
Note that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq84_HTML.gif . If not, then the above inequality gives
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equo_HTML.gif
a contradiction. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq85_HTML.gif . Continuing this process, we construct a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq86_HTML.gif in X such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq87_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq88_HTML.gif with
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equp_HTML.gif
By the given assumption, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq89_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq83_HTML.gif , so
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equq_HTML.gif
Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equr_HTML.gif
We claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq90_HTML.gif . If not, then by the above inequality, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equs_HTML.gif
a contradiction as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq91_HTML.gif . So, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equt_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ3_HTML.gif
(3)
Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq86_HTML.gif is a Cauchy sequence in X. Since X is complete, there is some point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq92_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq93_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq58_HTML.gif for all n, then by the assumption, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq56_HTML.gif . Now, we show that for every pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq94_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq95_HTML.gif , the following inequality holds:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equu_HTML.gif
As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq93_HTML.gif , there exists a positive integer http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq96_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq97_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ4_HTML.gif
(4)
Now, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq98_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equv_HTML.gif
implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equw_HTML.gif
which on taking limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq99_HTML.gif gives
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equx_HTML.gif
If
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equy_HTML.gif
then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equz_HTML.gif
Hence,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ5_HTML.gif
(5)
Now, we show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq100_HTML.gif for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq101_HTML.gif . First, consider the case http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq102_HTML.gif . Assume on the contrary that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq103_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq104_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq105_HTML.gif , as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq106_HTML.gif is nonempty and compact, so for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq107_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ6_HTML.gif
(6)
Now, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq105_HTML.gif implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq108_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq109_HTML.gif . From (5) we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ7_HTML.gif
(7)
Now,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaa_HTML.gif
implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equab_HTML.gif
Hence,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equac_HTML.gif
which further implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equad_HTML.gif
We claim that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq110_HTML.gif . If not, then the above inequality becomes
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equae_HTML.gif
a contradiction, so we deduce that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq111_HTML.gif . From inequality (7), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaf_HTML.gif
Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equag_HTML.gif

a contradiction. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq100_HTML.gif .

Now, when http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq112_HTML.gif , we first prove that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ8_HTML.gif
(8)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq94_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq113_HTML.gif , then (8) holds trivially. So, assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq95_HTML.gif . For every http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq114_HTML.gif , one may find a sequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq115_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equah_HTML.gif
As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq115_HTML.gif , this implies http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq116_HTML.gif . Using (7) we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equai_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq114_HTML.gif . If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq117_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaj_HTML.gif
This implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equak_HTML.gif
Hence, for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq118_HTML.gif , we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equal_HTML.gif
On taking the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq99_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equam_HTML.gif
If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq119_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equan_HTML.gif
On taking the limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq99_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equao_HTML.gif
By the given assumption, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equap_HTML.gif
Thus, for any http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq95_HTML.gif , (8) holds true. Put http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq120_HTML.gif in the above inequality to obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaq_HTML.gif

as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq121_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq122_HTML.gif . Hence by Lemma 2, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq100_HTML.gif . □

Corollary 5 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq51_HTML.gif be a complete ordered metric space. If an ordered fuzzy mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq59_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equar_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equas_HTML.gif

Then there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq15_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq30_HTML.gif provided that X satisfies the order sequential limit property.

Corollary 6 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq51_HTML.gif be a complete ordered metric space. If an ordered fuzzy mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq59_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equat_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equau_HTML.gif

and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq123_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq124_HTML.gif . Then there exists a point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq15_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq30_HTML.gif provided that X satisfies the order sequential limit property.

3 An application

Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq34_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq35_HTML.gif . A pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif is said to be an ordered fuzzy hybrid pair if the following conditions are satisfied:
  1. (c)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq125_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq61_HTML.gif .

     
  2. (d)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif gives http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq63_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq127_HTML.gif .

     
  3. (e)

    http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq128_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif .

     
Theorem 7 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq51_HTML.gif be a complete ordered metric space. If an ordered fuzzy hybrid pair http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq36_HTML.gif satisfies
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ9_HTML.gif
(9)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif , where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equav_HTML.gif
Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq129_HTML.gif provided that X satisfies the order sequential limit property and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq130_HTML.gif for each α. Moreover, F and g have a common fixed fuzzy point if any of the following conditions holds:
  1. (f)

    F and g are w-fuzzy compatible, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq132_HTML.gif for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq133_HTML.gif and g is continuous at u.

     
  2. (g)

    g is F-fuzzy weakly commuting for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq134_HTML.gif and is a fixed point of g, that is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq135_HTML.gif .

     
  3. (h)

    g is continuous at x for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq134_HTML.gif and for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq133_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq136_HTML.gif .

     
Proof By Lemma 1, there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq48_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq50_HTML.gif is one-to-one and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq49_HTML.gif . Define a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq137_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ10_HTML.gif
(10)
As g is one-to-one on E, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq138_HTML.gif is well defined. Also,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equ11_HTML.gif
(11)
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq62_HTML.gif . Therefore,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaw_HTML.gif
for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq128_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq138_HTML.gif satisfies (2) and all the conditions of Theorem 4. Using Theorem 4 with a mapping http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq138_HTML.gif , it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq138_HTML.gif has a fixed fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq139_HTML.gif . Now, it is left to prove that F and g have a coincidence fuzzy point. Since http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq138_HTML.gif has a fixed fuzzy point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq140_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq141_HTML.gif . As http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq130_HTML.gif , so there exists http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq76_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq142_HTML.gif , thus it follows that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq143_HTML.gif . This implies that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq76_HTML.gif is a coincidence fuzzy point of F and g. Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq129_HTML.gif . Suppose now that (f) holds. Then for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq144_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq145_HTML.gif , where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq133_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq146_HTML.gif . 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq147_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq148_HTML.gif . That is, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq149_HTML.gif for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq148_HTML.gif . Now,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equax_HTML.gif
implies that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equay_HTML.gif

On taking limit as http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq99_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq150_HTML.gif and therefore http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq151_HTML.gif . By Lemma 2 we obtain http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq152_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq153_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq154_HTML.gif is a common fixed fuzzy point of F and g. Suppose now that (g) holds. If for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq144_HTML.gif , g is F-fuzzy weakly commuting and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq135_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq155_HTML.gif . Hence, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq156_HTML.gif is a common fixed fuzzy point of F and g. Suppose now that (h) holds and assume that for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq144_HTML.gif and for some http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq133_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq157_HTML.gif . By the continuity of g at x and y, we get http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq41_HTML.gif . The result follows. □

Example 1 Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq158_HTML.gif be endowed with the usual metric. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq160_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq161_HTML.gif . Define a fuzzy mapping F from X into http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq162_HTML.gif as
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equaz_HTML.gif
and for http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq163_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equba_HTML.gif
Define a self-map http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq47_HTML.gif by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq164_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbb_HTML.gif
Note that for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbc_HTML.gif
Also, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq165_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbd_HTML.gif
And
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Eqube_HTML.gif
If http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq166_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq167_HTML.gif , then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq168_HTML.gif . So, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif , with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq169_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq170_HTML.gif . Hence, for all http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq53_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbf_HTML.gif
hold true, where
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbg_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_Equbh_HTML.gif

Hence, all the conditions of Theorem 7 are satisfied. Moreover, for each http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq171_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq172_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq173_HTML.gif . For http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq18_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2013-9/MediaObjects/13663_2012_366_IEq174_HTML.gif .

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 [2326]). 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.

Declarations

Acknowledgements

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

Authors’ Affiliations

(1)
Department of Mathematics, School of Science and Engineering, Lahore University of Management Sciences
(2)
Department of Mathematics and Applied Mathematics, University of Pretoria, Hatfield

References

  1. Heilpern S: Fuzzy mappings and fuzzy fixed point theorems. J. Math. Anal. Appl. 1981, 83:566–569.MathSciNetMATHView Article
  2. Sen CS: Fixed degree for fuzzy mappings and a generalization of Ky Fan’s theorem. Fuzzy Sets Syst. 1987, 24:103–112.View Article
  3. Estruch VD, Vidal A: A note on fixed fuzzy points for fuzzy mappings. Rend. Ist. Mat. Univ. Trieste 2001, 32:39–45.MathSciNetMATH
  4. Haghi RH, Rezapour S, Shahzad N: Some fixed point generalizations are not real generalizations. Nonlinear Anal. 2011, 74:1799–1803.MathSciNetMATHView Article
  5. Lee BS, Cho SJ: A fixed point theorem for contractive type fuzzy mappings. Fuzzy Sets Syst. 1994, 61:309–312.MathSciNetMATHView Article
  6. Zadeh LA: Fuzzy sets. Inf. Control 1965, 8:103–112.MathSciNetView Article
  7. Nadler SB Jr.: Multivalued contraction mappings. Pac. J. Math. 1969, 30:475–488.MathSciNetMATHView Article
  8. Sedghi S, Shobe N, Altun I: A fixed fuzzy point for fuzzy mappings in complete metric spaces. Math. Commun. 2008, 13:289–294.MathSciNetMATH
  9. Turkoglu D, Rhoades BE: A fixed fuzzy point for fuzzy mapping in complete metric spaces. Math. Commun. 2005, 10:115–121.MathSciNetMATH
  10. Suzuki T: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 2008, 136:1861–1869.MATHView Article
  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. 2012, 219:1712–1723.MathSciNetView Article
  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. 2004, 132:1435–1443.MathSciNetMATHView Article
  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. 2010, 72:2238–2242.MathSciNetMATHView Article
  18. Ćirić L, Abbas M, Saadati R, Hussain N: Common fixed points of almost generalized contractive mappings in ordered metric spaces. Appl. Math. Comput. 2011, 217:5784–5789.MathSciNetMATHView Article
  19. Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71:3403–3410.MathSciNetMATHView Article
  20. Kadelburg Z, Pavlović M, Radenović S: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl. 2010, 59:3148–3159.MathSciNetMATHView Article
  21. Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22:223–239.MathSciNetMATHView Article
  22. Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. 2010, 72:4508–4517.MathSciNetMATHView Article
  23. Baskaran R, Subrahmanyam PV: A note on the solution of a class of functional equations. Appl. Anal. 1986, 22:235–241.MathSciNetMATHView Article
  24. Bellman R: Methods of Nonlinear Analysis. Vol. II. Mathematics in Science and Engineering, vol. 61. Academic Press, New York (1973)
  25. Bellman R, Lee ES: Functional equations in dynamic programming. Aequ. Math. 1978, 17:1–18.MathSciNetMATHView Article
  26. Bhakta PC, Mitra S: Some existence theorems for functional equations arising in dynamic programming. J. Math. Anal. Appl. 1984, 98:348–362.MathSciNetMATHView Article

Copyright

© Ali and Abbas; licensee Springer 2013

This article is published under license to BioMed Central Ltd. 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.