Let

*X* be a space of points with generic elements of

*X* denoted by

*x* and

. 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

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

, is defined as

for

and for

, we have

where

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

,

is a compact, convex subset of

*X* and

Let
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
(that is, *B* includes *A*) if and only if
for each *x* in *X*, where
and
denote the membership function of *A* and *B*, respectively. It is easy to see that if
, then
.

Corresponding to each

and

, the fuzzy point

of

*X* is the fuzzy set

given by

For

, we have

Let

be a collection of all fuzzy subsets of

*X* and

be a subcollection of all approximate quantities. For

and

, define

Note that
is a nondecreasing function of *α* and *D* is a metric on
. Let
. Define
. Let
be a metric space and *Y* be an arbitrary set. A mapping
is called a fuzzy mapping, that is,
for each *y* in *Y*. Thus, if we characterize a fuzzy set *Fy* in a metric space *X* by a membership function *Fy*, then
is the grade of membership of *x* in *Fy*. Therefore, a fuzzy mapping *F* is a fuzzy subset of
with a membership function
.

In a more general sense than that given in [1], a mapping
is a fuzzy mapping over *X*[2] and
is the fixed degree of *x* in
.

**Definition 1** ([3])

A fuzzy point
in *X* is called a fixed fuzzy point of the fuzzy mapping *F* if
, that is,
or
. That is, the fixed degree of *x* in *Fx* is at least *α*. If
, then *x* is a fixed point of a fuzzy mapping *F*.

Let
and
.

A fuzzy point
in *X* is called a coincidence fuzzy point of the hybrid pair
if
, that is,
or
. That is, the fixed degree of *gx* in *Fx* is at least *α*. A fuzzy point
in *X* is called a common fixed fuzzy point of the hybrid pair
if
, that is,
(the fixed degree of *x* and *gx* in *Fx* is the same and is at least *α*).

We denote by
and
the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair
, respectively.

A hybrid pair
is called *w*-*fuzzy compatible* if
whenever
.

A mapping *g* is called *F*-*fuzzy weakly commuting* at some point
if
.

**Lemma 1** ([4])

*Let*
*X*
*be a nonempty set and*
. *Then there exists a subset*
*such that*
*and*
*is one*-*to*-*one*.

**Definition 2** Let *X* be a nonempty set. Then
is called an ordered metric space if
is a metric space and
is partially ordered.

Let
be a partially ordered set. Then
are said to be comparable if
or
holds.

An ordered metric space is said to satisfy the order sequential limit property if
for all *n*, whenever a sequence
and
for all *n*.

A mapping

is said to be an ordered fuzzy mapping if the following conditions are satisfied:

- (a)
implies that
.

- (b)

The following lemmas are needed in the sequel.

**Lemma 2** (Heilpern [1])

*Let*
*be a metric space*,

*and*
:

- 1.
*if*
, *then*
;

- 2.
;

- 3.
*if*
, *then*
.

**Lemma 3** (Lee and Cho [5])

*Let*
*be a complete metric space and*
*F*
*be a fuzzy mapping from*
*X*
*into*
*and*
. *Then there exists an*
*such that*
.

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

be the nonincreasing function defined by