Let

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

*X* denoted by

*x* and

$I=[0,1]$. 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

$(X,d)$ 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

${A}_{\alpha}$, is defined as

${A}_{\alpha}=\{x:A(x)\ge \alpha \}$

for

$\alpha \in (0,1]$ and for

$\alpha =0$, we have

${A}_{0}=\overline{\{x:A(x)>0\}},$

where

$\overline{B}$ 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

$\alpha \in [0,1]$,

${A}_{\alpha}$ is a compact, convex subset of

*X* and

$\underset{x\in X}{sup}A(x)=1.$

Let $W(X)$ 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 $A\subset B$ (that is, *B* includes *A*) if and only if $A(x)\le B(x)$ for each *x* in *X*, where $A(x)$ and $B(x)$ denote the membership function of *A* and *B*, respectively. It is easy to see that if $0<\alpha \le \beta \le 1$, then ${A}_{\alpha}\subseteq {A}_{\beta}$.

Corresponding to each

$\alpha \in [0,1]$ and

$x\in X$, the fuzzy point

${x}_{\alpha}$ of

*X* is the fuzzy set

${x}_{\alpha}:X\to [0,1]$ given by

${x}_{\alpha}(y)=\{\begin{array}{cc}\alpha \hfill & \text{if}x=y,\hfill \\ 0\hfill & \text{otherwise}.\hfill \end{array}$

For

$\alpha =1$, we have

${x}_{1}(y)=\{\begin{array}{cc}1\hfill & \text{if}x=y,\hfill \\ 0\hfill & \text{otherwise}\hfill \end{array}=\{x\}.$

Let

${I}^{X}$ be a collection of all fuzzy subsets of

*X* and

$W(X)$ be a subcollection of all approximate quantities. For

$A,B\in W(X)$ and

$\alpha \in [0,1]$, define

and

$D(A,B)=\underset{\alpha}{sup}{D}_{\alpha}(A,B).$

Note that ${p}_{\alpha}$ is a nondecreasing function of *α* and *D* is a metric on $W(X)$. Let $\alpha \in [0,1]$. Define ${W}_{\alpha}(X)=\{A\in {I}^{X}:{A}_{\alpha}\text{is nonempty, convex and compact}\}$. Let $(X,d)$ be a metric space and *Y* be an arbitrary set. A mapping $F:Y\to {W}_{\alpha}(X)$ is called a fuzzy mapping, that is, $Fy\in {W}_{\alpha}(X)$ for each *y* in *Y*. Thus, if we characterize a fuzzy set *Fy* in a metric space *X* by a membership function *Fy*, then $Fy(x)$ is the grade of membership of *x* in *Fy*. Therefore, a fuzzy mapping *F* is a fuzzy subset of $Y\times X$ with a membership function $Fy(x)$.

In a more general sense than that given in [1], a mapping $F:X\to {I}^{X}$ is a fuzzy mapping over *X*[2] and $(F(x)x)$ is the fixed degree of *x* in $F(x)$.

**Definition 1** ([3])

A fuzzy point ${x}_{\alpha}$ in *X* is called a fixed fuzzy point of the fuzzy mapping *F* if ${x}_{\alpha}\subset Fx$, that is, $(Fx)x\ge \alpha $ or $x\in {(Fx)}_{\alpha}$. That is, the fixed degree of *x* in *Fx* is at least *α*. If $\{x\}\subset Fx$, then *x* is a fixed point of a fuzzy mapping *F*.

Let $F:X\to {W}_{\alpha}(X)$ and $g:X\to X$.

A fuzzy point ${x}_{\alpha}$ in *X* is called a coincidence fuzzy point of the hybrid pair $\{F,g\}$ if ${(gx)}_{\alpha}\subset Fx$, that is, $(Fx)gx\ge \alpha $ or $gx\in {(Fx)}_{\alpha}$. That is, the fixed degree of *gx* in *Fx* is at least *α*. A fuzzy point ${x}_{\alpha}$ in *X* is called a common fixed fuzzy point of the hybrid pair $\{F,g\}$ if ${x}_{\alpha}={(gx)}_{\alpha}\subset Fx$, that is, $x=gx\in {(Fx)}_{\alpha}$ (the fixed degree of *x* and *gx* in *Fx* is the same and is at least *α*).

We denote by ${C}_{\alpha}(F,g)$ and ${F}_{\alpha}(F,g)$ the set of all coincidence fuzzy points and the set of all common fixed fuzzy points of the hybrid pair $\{F,g\}$, respectively.

A hybrid pair $\{F,g\}$ is called *w*-*fuzzy compatible* if $g{(Fx)}_{\alpha}\subseteq {(Fgx)}_{\alpha}$ whenever $x\in {C}_{\alpha}(F,g)$.

A mapping *g* is called *F*-*fuzzy weakly commuting* at some point $x\in X$ if ${g}^{2}(x)\in {(Fgx)}_{\alpha}$.

**Lemma 1** ([4])

*Let* *X* *be a nonempty set and*$g:X\to X$. *Then there exists a subset*$E\subseteq X$*such that*$g(E)=g(X)$*and*$g:E\to X$*is one*-*to*-*one*.

**Definition 2** Let *X* be a nonempty set. Then $(X,d,\le )$ is called an ordered metric space if $(X,d)$ is a metric space and $(X,\le )$ is partially ordered.

Let $(X,\le )$ be a partially ordered set. Then $x,y\in X$ are said to be comparable if $x\le y$ or $y\le x$ holds.

Define

$\mathrm{\nabla}=\{(x,y)\in X\times X:x\le y\text{or}y\le x\}.$

An ordered metric space is said to satisfy the order sequential limit property if $({u}_{n},z)\in \mathrm{\nabla}$ for all *n*, whenever a sequence ${u}_{n}\to z$ and $({u}_{n},{u}_{n+1})\in \mathrm{\nabla}$ for all *n*.

A mapping

$F:X\to {W}_{\alpha}(X)$ is said to be an ordered fuzzy mapping if the following conditions are satisfied:

- (a)
$y\in F{(x)}_{\alpha}$ implies that $(y,x)\in \mathrm{\nabla}$.

- (b)
$(x,y)\in \mathrm{\nabla}$ implies that $(u,v)\in \mathrm{\nabla}$ whenever $u\in {(Fx)}_{\alpha}$ and $v\in {(Fy)}_{\alpha}$.

The following lemmas are needed in the sequel.

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

*Let*$(X,d)$*be a metric space*,

$x,y\in X$*and*$A,B\in W(X)$:

- 1.
*if* ${p}_{\alpha}(x,A)=0$, *then* ${x}_{\alpha}\subset A$;

- 2.
${p}_{\alpha}(x,A)\le d(x,y)+{p}_{\alpha}(y,A)$;

- 3.
*if* ${x}_{\alpha}\subset A$, *then* ${p}_{\alpha}(x,B)\le {D}_{\alpha}(A,B)$.

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

*Let*$(X,d)$*be a complete metric space and* *F* *be a fuzzy mapping from* *X* *into*$W(X)$*and*${x}_{0}\in X$. *Then there exists an*${x}_{1}\in X$*such that*$\{{x}_{1}\}\subset F{x}_{0}$.

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

$\sigma :[0,1)\to (0,1]$ be the nonincreasing function defined by

$\sigma (r)=\{\begin{array}{cc}1\hfill & \text{if}0\le r\frac{1}{2},\hfill \\ 1-r\hfill & \text{if}\frac{1}{2}\le r1.\hfill \end{array}$

(1)