#
*L*-fuzzy Fixed Point Theorems for *L*-fuzzy Mappings via \(\beta _{F_{L}}\)-admissible with Applications

- Muhammad Sirajo Abdullahi
^{1}Email authorView ORCID ID profile and - Akbar Azam
^{2}

**5**:2

https://doi.org/10.1186/s40467-017-0056-5

© The Author(s) 2017

**Received: **9 September 2016

**Accepted: **8 February 2017

**Published: **20 February 2017

## Abstract

In this paper, the authors use the idea of \(\beta _{F_{L}}\)-admissible mappings to prove some *L*-fuzzy fixed point theorems for a generalized contractive *L*-fuzzy mappings. Some examples and applications to *L*-fuzzy fixed points for *L*-fuzzy mappings in partially ordered metric spaces are also given, to support main findings.

## Keywords

*L*-fuzzy sets

*L*-fuzzy fixed points

*L*-fuzzy mappings \(\beta _{F_{L}}\)-admissible mappings

## AMS Subject Classification

## Introduction

Solving real-world problems becomes apparently easier with the introduction of fuzzy set theory in 1965 by L. A Zadeh [1], as it helps in making the description of vagueness and imprecision clear and more precise. Later in 1967, Goguen [2] extended this idea to *L*-fuzzy set theory by replacing the interval [ 0,1] with a completely distributive lattice *L*.

In 1981, Heilpern [3] gave a fuzzy extension of Banach contraction principle [4] and Nadler’s [5] fixed point theorems by introducing the concept of fuzzy contraction mappings and established a fixed point theorem for fuzzy contraction mappings in a complete metric linear spaces. Frigon and Regan [6] generalized the Heilpern theorem under a contractive condition for 1-level sets of a fuzzy contraction on a complete metric space, where the 1-level sets need not be convex and compact. Subsequently, various generalizations of result in [6] were obtained (see [7–12]). While in 2001, Estruch and Vidal [13] established the existence of a fixed fuzzy point for fuzzy contraction mappings (in the Helpern’s sense) on a complete metric space. Afterwards, several authors [11, 14–17] among others studied and generalized the result in [13].

On the other hand, the concept of *β*-admissible mapping was introduced by Samet et al. [18] for a single-valued mappings and proved the existence of fixed point theorems via this concept, while Asl et al. [19] extended the notion to *α*−*ψ*-multi-valued mappings. Afterwards, Mohammadi et al. [20] established the notion of *β*-admissible mapping for the multi-valued mappings (different from the *β*
_{∗}-admissible mapping provided in [19]).

Recently, Phiangsungnoen et al. [21] use the concept of *β*-admissible defined by Mohammadi et al. [20] to proved some fuzzy fixed point theorems. In 2014, Rashid et al. [22] introduced the notion of \(\beta _{F_{L}}\)-admissible for a pair of *L*-fuzzy mappings and utilized it to proved a common *L*-fuzzy fixed point theorem. The notions of \(d_{L}^{\infty }\)-metric and Hausdorff distances for *L*-fuzzy sets were introduced by Rashid et al. [23], they presented some fixed point theorems for *L*-fuzzy set valued-mappings and coincidence theorems for a crisp mapping and a sequence of *L*-fuzzy mappings. Many researchers have studied fixed point theory in the fuzzy context of metric spaces and normed spaces (see [24–27] and [28–30], respectively).

In this manuscript, the authors developed a new *L*-fuzzy fixed point theorems on a complete metric space via \(\beta _{F_{L}}\)-admissble mapping in sense of Mohammadi et al. [20] which is a generalization of main result of Phiangsungnoen et al. [21]. We also construct some examples to support our results and infer as an application, the existence of *L*-fuzzy fixed points in a complete partially ordered metric space.

## Preliminaries

In this section we present some basic definitions and preliminary results which we will used throughout this paper. Let (*X*,*d*) be a metric space, *C*
*B*(*X*)={*A*:*A* is closed and bounded subsets of *X*} and *C*(*X*)={*A*:*A* is nonempty compactsubsets of *X*}.

*A*,

*B*∈

*C*

*B*(

*X*) and define

###
**Definition 1**

A fuzzy set in *X* is a function with domain *X* and range in [ 0,1]. i.e *A* is a fuzzy set if *A*:*X*→[ 0,1].

Let \({\mathcal F}(X)\) denotes the collection of all fuzzy subsets of *X*. If *A* is a fuzzy set and *x*∈*X*, then *A*(*x*) is called the grade of membership of *x* in *A*. The *α*-level set of *A* is denoted by [ *A*]_{
α
} and is defined as below:

[ *A*]_{
α
}={*x*∈*X*:*A*(*x*)≥*α*},for *α*∈(0,1],

[ *A*]_{0}=closure of the set {*x*∈*X*:*A*(*x*)>0}.

###
**Definition 2**

*L*,≼

_{ L }) is called

- i
a lattice; if

*a*∨*b*∈*L*,*a*∧*b*∈*L*for any*a*,*b*∈*L*, - ii
a complete lattice; if \(\bigvee A\in L, \bigwedge A \in L\ \text {for any}\ A \subseteq L,\)

- iii
a distributive lattice; if

*a*∨(*b*∧*c*)=(*a*∨*b*)∧(*a*∨*c*),*a*∧(*b*∨*c*)=(*a*∧*b*)∨(*a*∧*c*) for any*a*,*b*,*c*∈*L*, - iv
a complete distributive lattice; if \(a\vee (\bigwedge b_{i}) = \bigwedge _{i} (a \wedge b_{i}), {\newline } a \wedge (\bigvee _{i} b_{i}) = \bigvee _{i} (a \wedge b_{i})\ \text {for any}\ a,b_{i} \in L,\)

- v
a bounded lattice; if it is a lattice and additionally has a top element 1

_{ L }and a bottom element 0_{ L }, which satisfy 0_{ L }≼_{ L }*x*≼_{ L }1_{ L }for every*x*∈*L*.

###
**Definition 3**

An *L*-fuzzy set *A* on a nonempty set *X* is a function *A*:*X*→*L*, where *L* is bounded complete distributive lattice with 1_{
L
} and 0_{
L
}.

###
**Definition 4**

*(Goguen [*
2
*])*. Let *L* be a lattice, the top and bottom elements of *L* are 1_{
L
} and 0_{
L
} respectively, and if *a*,*b*∈*L*,*a*∨*b*=1_{
L
} and *a*∧*b*=0_{
L
} then *b* is a unique complement of \(a\ \text {denoted by}\ \acute {a}\).

###
**Remark 1**

If *L*=[ 0,1], then the *L*-fuzzy set is the special case of fuzzy sets in the original sense of *Zadeh [*
1
*]*, which shows that *L*-fuzzy set is larger.

*L*-fuzzy subsets of

*X*. Define \(\mathcal Q_{L}(X) \subset \mathcal F_{L}(X)\) as below:

The *α*
_{
L
}-level set of an *L*-fuzzy set *A* is denoted by \(A_{\alpha _{L}}\) and define as below:

\(A_{\alpha _{L}} = \{x\in X : \alpha _{L} \preceq _{L} A(x)\}\ \text {for}\ \alpha _{L}\in L\backslash \{0_{L}\}\),

\(A_{0_{L}} = \overline {\{x\in X : 0_{L} \preceq _{L} A(x)\}}\).

Where \(\overline {B}\) denotes the closure of the set *B* (Crisp).

*A*⊂

*B*if and only if

*A*(

*x*)≼

_{ L }

*B*(

*x*) for all

*x*∈

*X*. If there exists an

*α*

_{ L }∈

*L*∖{0

_{ L }} such that \(A_{\alpha _{L}}, B_{\alpha _{L}}\in CB(X),\) then we define

We note that \(d_{L}^{\infty }\) is a metric on \(\mathcal F_{L}(X)\) and the completeness of (*X*,*d*) implies that (*C*(*X*),*H*) and \((\mathcal F_{L}(X), d_{L}^{\infty })\) are complete.

###
**Definition 5**

Let *X* be an arbitrary set, *Y* be a metric space. A mapping *T* is called *L*-fuzzy mapping, if *T* is a mapping from *X* to \({\mathcal F}_{L}(Y)\)(i.e class of L-fuzzy subsets of *Y*). An *L*-fuzzy mapping *T* is an *L*-fuzzy subset on *X*×*Y* with membership function *T*(*x*)(*y*). The function *T*(*x*)(*y*) is the grade of membership of *y* in *T*(*x*).

###
**Definition 6**

*X*be a nonempty set. For

*x*∈

*X*, we write {

*x*} the characteristic function of the ordinary subset {

*x*} of

*X*. The characteristic function of an

*L*-fuzzy set

*A*, is denoted by \(\chi _{L_{A}}\) and define as below:

###
**Definition 7**

Let (*X*,*d*) be a metric space and \(T: X\longrightarrow {\mathcal F}_{L}(X)\). A point *z*∈*X* is said to be an *L*-fuzzy fixed point of *T* if \(z\in \, [\!Tz]_{\alpha _{L}}\), for some *α*
_{
L
}∈*L*∖{0_{
L
}}.

###
**Remark 2**

If *α*
_{
L
}=1_{
L
}, then it is called a fixed point of the *L*-fuzzy mapping *T*.

###
**Definition 8**

*X*be a nonempty set.

*T*:

*X*→2

^{ X }, where 2

^{ X }is a collection of nonempty subsets of

*X*and

*β*:

*X*×

*X*→[ 0,

*∞*). We say that

*T*is

*β*

_{∗}-admissible if

###
**Definition 9**

(Mohammadi et al. [20]). Let *X* be a nonempty set. *T*:*X*→2^{
X
}, where 2^{
X
} is a collection of nonempty subsets of *X* and *β*:*X*×*X*→[ 0,*∞*). We say that *T* is *β*-admissible whenever for each *x*∈*X* and *y*∈*T*
*x* with *β*(*x*,*y*)≥1, we have *β*(*y*,*z*)≥1 for all *z*∈*T*
*y*.

###
**Remark 3**

If *T* is *β*
_{∗}-admissible mapping, then T is also *β*-admissible mapping.

###
**Example 1**

*X*=[ 0,

*∞*) and

*d*(

*x*,

*y*)=|

*x*−

*y*|. Define

*T*:

*X*→2

^{ X }and

*β*:

*X*×

*X*→[ 0,

*∞*) by

*T*is

*β*-admissible.

## Main Result

##
*L*-fuzzy Fixed Point Theorems

Now, we recall some well known results and definitions to be used in the sequel.

###
**Lemma 1**

Let \(x\in X, A\in {\mathcal W}_{L}(X), \text {and}\ \{x\}\) be an L-fuzzy set with membership function equal to characteristic function of set \(\{x\}.\ \text {If}\ \{x\} \subset A,\ \text {then}\ p_{\alpha _{L}}(x,A) = 0\ \text {for}\ \alpha _{L}\in \ L\backslash \{0_{L}\}\).

###
**Lemma 2**

(Nadler [5]). Let (*X*,*d*) be a metric space and *A*,*B*∈*C*
*B*(*X*). Then for any *a*∈*A* there exists *b*∈*B* such that *d*(*a*,*b*)≤*H*(*A*,*B*).

###
**Definition 10**

Let *Ψ* be the family of non-decreasing functions *ψ*:[ 0,*∞*)→[ 0,*∞*) such that \(\sum ^{\infty }_{n=1} \psi ^{n}(t) < \infty \) for all *t*>0 where *ψ*
^{
n
} is the nth iterate of *ψ*. It is known that *ψ*(*t*)<*t* for all *t*>0 and *ψ*(0)=0.

Below, we introduce the concept of *β*-admissible in the sense of Mohammadi et al. [20] for *L*-fuzzy mappings.

###
**Definition 11**

Let (*X*,*d*) be a metric space, *β*:*X*×*X*→[ 0,*∞*) and *T*:*X*→*F*
_{
L
}(*X*). A mapping *T* is said to be \(\beta _{F_{L}}\)-admissible whenever for each *x*∈*X* and \(y \in \, [\!Tx]_{\alpha _{L}}\) with *β*(*x*,*y*)≥1, we have *β*(*y*,*z*)≥1 for all \(z \in \, [\!Ty]_{\alpha _{L}}\), where *α*
_{
L
}∈*L*∖{0_{
L
}}.

Here, the existence of an *L*-fuzzy fixed point theorem for some generalized type of contraction *L*-fuzzy mappings in complete metric spaces is presented.

###
**Theorem 1**

*X*,

*d*)be a complete metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*and

*β*:

*X*×

*X*→[ 0,

*∞*) such that for all

*x*,

*y*∈

*X*,

*K*≥0 and

- i.
if {

*x*_{ n }} is a sequence in*X*so that*β*(*x*_{ n },*x*_{ n+1})≥1 and*x*_{ n }→*b*(*n*→*∞*), then*β*(*x*_{ n },*b*)≥1, - ii.
there exists

*x*_{0}∈*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) so that*β*(*x*_{0},*x*_{1})≥1, - iii.
*T*is \(\beta _{F_{L}}\)-admissible, - iv.
*ψ*is continuous.

Then *T* has atleast an *L*-fuzzy fixed point.

###
*Proof*

*x*

_{0}∈

*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) by condition (ii) we have

*β*(

*x*

_{0},

*x*

_{1})≥1. Since \([\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) is nonempty and compact, then there exists \(x_{2} \in \, [\!{Tx}_{1}]_{\alpha _{L}}\phantom {\dot {i}\!}\), such that

*β*(

*x*

_{0},

*x*

_{1})≥1, we have

*x*

_{2}∈

*X*, we have \([\!{Tx}_{2}]_{\alpha _{L}}\phantom {\dot {i}\!}\) which is nonempty and compact subset of

*X*, then there exists \(x_{3} \in \, [\!{Tx}_{2}]_{\alpha _{L}}\phantom {\dot {i}\!}\), such that

*x*

_{0}∈

*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) with

*β*(

*x*

_{0},

*x*

_{1})≥1, by condition (iii) we have

*β*(

*x*

_{1},

*x*

_{2})≥1. From (1), (2) and the fact that

*β*(

*x*

_{1},

*x*

_{2})≥1, we have

*x*

_{ n }} is obtained such that, for each

*n*∈

**N**, \(x_{n} \in [\!{Tx}_{n-1}]_{\alpha _{L}}\phantom {\dot {i}\!}\) with

*β*(

*x*

_{ n−1},

*x*

_{ n })≥1, we have

*L*-fuzzy fixed point of

*T*. So, we suppose that for each \(n \in \mathbb N\), \(p_{\alpha _{L}}(x_{n}, {Tx}_{n}) > 0\), implying that

*d*(

*x*

_{ n−1},

*x*

_{ n })>0 for all \(n \in \mathbb N\). Thus, if

*d*(

*x*

_{ n },

*x*

_{ n+1})>

*d*(

*x*

_{ n−1},

*x*

_{ n }) for some \(n \in \mathbb N\), then by (4) and Definition 10, we have

*x*

_{ n }} is a Cauchy sequence in

*X*. Since

*ψ*∈

*Ψ*and continuous, then there exist

*ε*>0 and a positive integer

*h*=

*h*(

*ε*) such that

*m*>

*n*>

*h*. By triangular inequality, (5) and (6), we have

*x*

_{ n }} is Cauchy sequence and since

*X*is complete therefore we have

*b*∈

*X*so that

*x*

_{ n }→

*b*as

*n*→

*∞*. Now, we show that \(b \in [\!Tb]_{\alpha _{L}}\phantom {\dot {i}\!}\). Let us assume the contrary and consider

*n*→

*∞*in (7), we have

Next, we give an example to support the validity of our result.

###
**Example 2**

*X*=[ 0,1],

*d*(

*x*,

*y*)=|

*x*−

*y*| for all

*x*,

*y*∈

*X*, then (

*X*,

*d*) is a complete metric space. Let

*L*={

*η*,

*κ*,

*ω*,

*τ*} with

*η*≼

_{ L }

*κ*≼

_{ L }

*τ*, and

*η*≼

_{ L }

*ω*≼

_{ L }

*τ*, where

*κ*and

*ω*are not comparable, therefore (

*L*,≼

_{ L })is a complete distributive lattice. Define \(T: X \longrightarrow \mathcal Q_{L}(X)\) as below:

*x*∈

*X*,

*α*

_{ L }=

*τ*exists for which

*β*:

*X*×

*X*→[ 0,

*∞*) as below:

*T*is \(\beta _{F_{L}}\)-admissible. For each

*x*,

*y*∈

*X*we have

Where \(\psi (t) = \frac {t}{3}\) for all *t*>0 and *K*≥0. Conditions (ii) and (iii) of Theorem 1 holds obviously. Thus, all the conditions of Theorem 1 are satisfied. Hence, there exists a 0∈*X* such that 0∈ [ *T*0]_{
τ
}.

Below, we introduce the concept of *β*
_{∗}-admissible for *L*-fuzzy mappings in the sense of Asl et al. [19].

###
**Definition 12**

*X*,

*d*)be a metric space,

*β*:

*X*×

*X*→[ 0,

*∞*) and

*T*:

*X*→

*F*

_{ L }(

*X*). A mapping

*T*is said to be \(\beta _{F_{L}}^{*}\)-admissible if

###
**Theorem 2**

*X*,

*d*)be a complete metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*and

*β*:

*X*×

*X*→[ 0,

*∞*) such that for all

*x*,

*y*∈

*X*,

*K*≥0 and

- i.
if {

*x*_{ n }} is a sequence in*X*such that*β*(*x*_{ n },*x*_{ n+1})≥1 and*x*_{ n }→*u*as*n*→*∞*, then*β*(*x*_{ n },*u*)≥1, - ii.
there exist

*x*_{0}∈*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that*β*(*x*_{0},*x*_{1})≥1, - iii.
*T*is \(\beta _{F_{L}}^{*}\)-admissible, - iv.
*ψ*is continuous.

Then, *T* has atleast an *L*-fuzzy fixed point.

###
*Proof*

By Remark 3 and Theorem 1 the result follows immediately. □

Taking *K*=0 in Theorem 1 and 2, we obtain the following corollary.

###
**Corollary 1**

*X*,

*d*)be a complete metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*and

*β*:

*X*×

*X*→[ 0,

*∞*) such that for all

*x*,

*y*∈

*X*,

- i.
if {

*x*_{ n }} is a sequence in*X*such that*β*(*x*_{ n },*x*_{ n+1})≥1 and*x*_{ n }→*u*as*n*→*∞*, then*β*(*x*_{ n },*u*)≥1, - ii.
there exist

*x*_{0}∈*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that*β*(*x*_{0},*x*_{1})≥1, - iii.
*T*is \(\beta _{F_{L}}\)-admissible (or \(\beta ^{*}_{F_{L}}\)-admissible), - iv.
*ψ*is continuous.

Then, *T* has atleast an *L*-fuzzy fixed point.

If *β*(*x*,*y*)=1 for all *x*,*y*∈*X*. Theorem 1 or 2 will reduce to the following result.

###
**Corollary 2**

*X*,

*d*)be a complete metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*such that for all

*x*,

*y*∈

*X*,

*K*≥0 and

Then, *T* has atleast an *L*-fuzzy fixed point.

By taking *K*=0 and *β*(*x*,*y*)=1 for all *x*,*y*∈*X* in Theorem 1 or 2, Corollary 1 or 2, we have the following.

###
**Corollary 3**

*X*,

*d*)be a complete metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*such that for all

*x*,

*y*∈

*X*,

Then, *T* has atleast an *L*-fuzzy fixed point.

## Applications

In this section, we establish as an application the existence of an *L*-fuzzy fixed point theorems in complete partially ordered metric spaces.

Below, we present some results which are essential in the remaining part of our work.

###
**Definition 13**

Let *X* be a nonempty set. Then, (*X*,*d*,≼) is said to be an ordered metric space if (*X*,*d*) is a metric space and (*X*,≼) is a partially ordered set.

###
**Definition 14**

Let (*X*,≼) be a partially ordered set. Then, *x*,*y*∈*X* are said to be comparable if *x*≼*y* or *y*≼*x* holds.

*X*,≼), we define

###
**Definition 15**

A partially ordered set (*X*,≼) is said to satisfy the ordered sequential limit property if \((x_{n}, x) \in \barwedge \) for all \(n \in \mathbb {N},\) whenever a sequence *x*
_{
n
}→*x* as *x*→*∞* and \((x_{n}, x_{n+1}) \in \barwedge \) for all \(n \in \mathbb {N}\).

###
**Definition 16**

Let (*X*,≼) be a partially ordered set and *α*
_{
L
}∈*L*∖{0_{
L
}}. An *L*-fuzzy mapping \(T: X \longrightarrow \mathcal Q_{L}(X)\) is said to be comparative, if for each *x*∈*X* and \(y \in \, [\!Tx]_{\alpha _{L}}\phantom {\dot {i}\!}\) with \((x, y) \in \barwedge,\) we have \((y, z) \in \barwedge \) for all \(z \in \, [\!Ty]_{\alpha _{L}}\phantom {\dot {i}\!}\).

Now, the existence of an *L*-fuzzy fixed point theorem for *L*-fuzzy mappings in complete partially ordered metric spaces is presented.

###
**Theorem 3**

*X*,

*d*,≼)be a complete partially ordered metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*such that for all \((x,y) \in \barwedge,\)

*K*≥0 and

- I.
*X*satisfies the order sequential limit property, - II.
there exist

*x*_{0}∈*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that \((x_{0}, x_{1}) \in \barwedge,\) - III.
*T*is comparative*L*-fuzzy mapping, - IV.
*ψ*is continuous.

Then, *T* has atleast an *L*-fuzzy fixed point.

###
*Proof*

*β*:

*X*×

*X*→[ 0,

*∞*) be defined as:

*β*(

*x*

_{0},

*x*

_{1})≥1 which implies that condition (ii) of Theorem 1 holds. And since

*T*is comparative

*L*-fuzzy mapping, then condition (iii) of Theorem 1 follows. By (8) and for all

*x*,

*y*∈

*X*, we have

Condition (i) of Theorem 1 also holds by condition (I). Now that all the hypothesis of Theorem 1 are fulfilled, hence the existence of the *L*-fuzzy fixed point for *L*-fuzzy mapping *T* follows. □

Applying similar technique in the proof of Theorem 3 with Corollary 1, we arrive at the following result.

###
**Corollary 4**

*X*,

*d*,≼)be a complete partially ordered metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*such that for all \((x,y) \in \barwedge,\)

- I.
*X*satisfies the order sequential limit property, - II.
there exist

*x*_{0}∈*X*and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that \((x_{0}, x_{1}) \in \barwedge,\) - III.
*T*is comparative*L*-fuzzy mapping, - IV.
*ψ*is continuous.

Then, *T* has at least an *L*-fuzzy fixed point.

Setting *β*(*x*,*y*)=1 for all \((x,y) \in \barwedge \) and using similar argument in the proof of Theorem 3 with Corollary 2 and 3 we get the followings, respectively.

###
**Corollary 5**

*X*,

*d*,≼)be a complete partially ordered metric space,

*α*

_{ L }∈

*L*∖{0

_{ L }} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an

*L*-fuzzy mapping. Suppose that there exist

*ψ*∈

*Ψ*such that for all \((x,y) \in \barwedge,\)

*K*≥0 and

Then, *T* has at least an *L*-fuzzy fixed point.

###
**Corollary 6**

Let (*X*,*d*,≼) be a complete partially ordered metric space, *α*
_{
L
}∈*L*∖{0_{
L
}} and \(T: X \longrightarrow \mathcal Q_{L}(X)\) be an *L*-fuzzy mapping. Suppose that there exist *ψ*∈*Ψ* such that for all \((x,y) \in \barwedge,\)

*T*has at least an

*L*-fuzzy fixed point.

## Declarations

### Acknowledgements

The authors thank the Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan for providing excellent research facilities.

### Authors’ contributions

Both authors contributed to the writing of this paper. Both authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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