Open Access

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

Journal of Uncertainty Analysis and Applications20175:2

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

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

Primary 46S40Secondary 47H1054H25

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 [712]). 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, 1417] 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 αψ-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 [2427] and [2830], 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}.

Let A,BC B(X) and define
$$\begin{aligned} d(x, A) &= \inf_{\substack{y \in A}} d(x, y),\\[-1pt] d(A, B) &= \inf_{\substack{x \in A, y \in B}} d(x, y),\\[-1pt] p_{\alpha_{L}}(x, A) &= \inf_{\substack{y \in A_{\alpha_{L}}}} d(x, y),\\[-1pt] p_{\alpha_{L}}(A, B) &= \inf_{\substack{x \in A_{\alpha_{L}}, y \in B_{\alpha_{L}}}} d(x, y),\\[-1pt] p(A, B) &= \sup_{\substack{\alpha_{L}}} p_{\alpha_{L}} (A, B),\\[-1pt] H\left(A_{\alpha_{L}}, B_{\alpha_{L}}\right) &= \max \bigg\{\sup_{\substack{x \in A_{\alpha_{L}}}} d\left(x, B_{\alpha_{L}}\right), \sup_{\substack{y \in B_{\alpha_{L}}}} d\left(y, A_{\alpha_{L}}\right)\bigg\},\\[-1pt] D_{\alpha_{L}}(A, B) &= H\left(A_{\alpha_{L}}, B_{\alpha_{L}}\right),\\[-1pt] d_{\alpha_{L}}^{\infty} (A, B) &= \sup_{\substack{\alpha_{L}}} D_{\alpha_{L}} (A, B). \end{aligned} $$

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 xX, 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 ={xX:A(x)≥α},for α(0,1],

[ A]0=closure of the set {xX:A(x)>0}.

Definition 2

A partially ordered set (L, L ) is called
  1. i

    a lattice; if abL,abL for any a,bL,

     
  2. ii

    a complete lattice; if \(\bigvee A\in L, \bigwedge A \in L\ \text {for any}\ A \subseteq L,\)

     
  3. iii

    a distributive lattice; if a(bc)=(ab)(ac),a(bc)=(ab)(ac) for any a,b,cL,

     
  4. iv

    a complete distributive lattice; if \(a\vee (\bigwedge b_{i}) = \bigwedge _{i} (a \wedge b_{i}), {\newline } a \wedge (\bigvee _{i} b_{i}) = \bigvee _{i} (a \wedge b_{i})\ \text {for any}\ a,b_{i} \in L,\)

     
  5. 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 xL.

     

Definition 3

An L-fuzzy set A on a nonempty set X is a function A:XL, 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,bL,ab=1 L and ab=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.

Let \({\mathcal F}_{L}(X)\) denotes the class of all L-fuzzy subsets of X. Define \(\mathcal Q_{L}(X) \subset \mathcal F_{L}(X)\) as below:
$$\mathcal Q_{L}(X) = \{A \in \mathcal F_{L}(X) : A_{\alpha_{L}}\ \text{is nonempty and compact,}\ \alpha_{L}\in L\backslash\{0_{L}\}\}. $$

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).

For \(A,B\in {\mathcal F}_{L}(X)\), AB if and only if A(x) L B(x) for all xX. If there exists an α L L{0 L } such that \(A_{\alpha _{L}}, B_{\alpha _{L}}\in CB(X),\) then we define
$$D_{\alpha_{L}}(A,B) = H(A_{\alpha_{L}}, B_{\alpha_{L}}). $$
If \(A_{\alpha _{L}}, B_{\alpha _{L}}\in CB(X)\ \text {for each}\ \alpha _{L}\in L\backslash \{0_{L}\}\), then we define
$$d^{\infty}_{L}(A,B) = \sup_{\alpha_{L}} D_{\alpha_{L}}(A,B). $$

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

Let X be a nonempty set. For xX, 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:
$$\chi_{L_{A}} = \left\{\begin{array}{ll} 0_{L} & \text{if}\ x \notin A;\\ 1_{L} & \text{if}\ x \in A. \end{array} \right.$$

Definition 7

Let (X,d) be a metric space and \(T: X\longrightarrow {\mathcal F}_{L}(X)\). A point zX 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

(Asl et al. [19]). 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 if
$$\text{for}\ x,y \in X, \beta(x, y) \geq 1 \Longrightarrow \beta_{*}(Tx, Ty) \geq 1, $$
where
$$\beta_{*}(Tx, Ty) := \inf{\{\beta(a,b) : a \in Tx\ \text{and}\ b\in Ty\}}. $$

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 xX and yT x with β(x,y)≥1, we have β(y,z)≥1 for all zT y.

Remark 3

If T is β -admissible mapping, then T is also β-admissible mapping.

Example 1

Let X=[ 0,) and d(x,y)=|xy|. Define T:X→2 X and β:X×X→[ 0,) by
$$T(x) = \left\{\begin{array}{ll} \left[0, \frac{x}{3}\right], & \text{if \(0 \leq x \leq 1\)};\\ \left[x^{2}\right., \left.\infty\right), & \text{if \(x > 1\)}. \end{array} \right.$$
and
$$\beta(x,y) = \left\{\begin{array}{ll} 1, & \text{if \(x,y \in\, [\!0, 1]\)};\\ 0, & \text{otherwise}. \end{array} \right.$$
Then, 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,BC B(X). Then for any aA there exists bB 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:XF L (X). A mapping T is said to be \(\beta _{F_{L}}\)-admissible whenever for each xX 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

Let (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 ψΨ and β:X×X→[ 0,) such that for all x,yX,
$$ \begin{aligned} \beta(x, y) D_{\alpha_{L}} (Tx, Ty) &\leq \psi (\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, \end{aligned} $$
(1)
where K≥0 and
$$\begin{array}{*{20}l} \Omega(x, y) = \max \biggl\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2} \biggr\}. \end{array} $$
If the following conditions hold,
  1. 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,

     
  2. ii.

    there exists x 0X and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) so that β(x 0,x 1)≥1,

     
  3. iii.

    T is \(\beta _{F_{L}}\)-admissible,

     
  4. iv.

    ψ is continuous.

     

Then T has atleast an L-fuzzy fixed point.

Proof

For x 0X 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
$$ d(x_{1},x_{2}) = p_{\alpha_{L}}(x_{1}, {Tx}_{1}) \leq D_{\alpha_{L}}({Tx}_{0},{Tx}_{1}). $$
(2)
By (2) and the fact that β(x 0,x 1)≥1, we have
$$\begin{array}{*{20}l} d(x_{1},x_{2}) & \leq D_{\alpha_{L}}({Tx}_{0},{Tx}_{1}) \\ & \leq \beta(x_{0}, x_{1}) D_{\alpha_{L}}({Tx}_{0},{Tx}_{1}) \\ & \leq \psi(\Omega(x_{0}, x_{1})) + K \min \left\{p_{\alpha_{L}}(x_{0}, {Tx}_{0}), p_{\alpha_{L}}(x_{1}, {Tx}_{1}),\right. \\ &\quad \left. p_{\alpha_{L}}(x_{0}, {Tx}_{1}), p_{\alpha_{L}}(x_{1}, {Tx}_{0})\right\} \\ & \leq \psi(\Omega(x_{0}, x_{1})) + K \min \left\{p_{\alpha_{L}}(x_{0}, x_{1}), p_{\alpha_{L}}(x_{1}, x_{2}), p_{\alpha_{L}}(x_{0}, x_{2}), 0\right\} \\ & = \psi(\Omega(x_{0}, x_{1})). \end{array} $$
Similarly, For x 2X, 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
$$ d(x_{2},x_{3}) = p_{\alpha_{L}}(x_{2}, {Tx}_{2}) \leq D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}). $$
(3)
For x 0X 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
$$\begin{array}{*{20}l} d(x_{2},x_{3}) & \leq D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}) \\ & \leq \beta(x_{1}, x_{2}) D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}) \\ & \leq \psi(\Omega(x_{1}, x_{2})) + K \min \left\{p_{\alpha_{L}}(x_{1}, {Tx}_{1}), p_{\alpha_{L}}(x_{2}, {Tx}_{2}),\right. \\ &\quad \left. p_{\alpha_{L}}(x_{1}, {Tx}_{2}), p_{\alpha_{L}}(x_{2}, {Tx}_{1})\right\} \\ & \leq \psi(\Omega(x_{1}, x_{2})) + K \min \left\{p_{\alpha_{L}}(x_{1}, x_{2}), p_{\alpha_{L}}(x_{2}, x_{3}), p_{\alpha_{L}}(x_{1}, x_{3}), 0\right\} \\ & = \psi(\Omega(x_{1}, x_{2})). \end{array} $$
Continuing in this pattern, a sequence {x n } is obtained such that, for each nN, \(x_{n} \in [\!{Tx}_{n-1}]_{\alpha _{L}}\phantom {\dot {i}\!}\) with β(x n−1,x n )≥1, we have
$$d\left(x_{n}, x_{n+1}\right) \leq \psi\left(\Omega\left(x_{n-1}, x_{n}\right)\right), $$
where
$$\begin{array}{*{20}l} \Omega\left(x_{n-1}, x_{n}\right) & = \max \bigg\{d\left(x_{n-1}, x_{n}\right), p_{\alpha_{L}}\left(x_{n-1}, {Tx}_{n-1}\right), \\ &\quad p_{\alpha_{L}}\left(x_{n}, {Tx}_{n}\right), \frac{p_{\alpha_{L}}\left(x_{n-1}, {Tx}_{n}\right) + p_{\alpha_{L}}\left(x_{n}, {Tx}_{n-1}\right)}{2}\bigg\} \\ & \leq \max \left\{d\left(x_{n-1}, x_{n}\right), d\left(x_{n}, x_{n+1}\right), \frac{d\left(x_{n-1}, x_{n+1}\right)}{2}\right\} \\ & = \max \{d\left(x_{n-1}, x_{n}\right), d\left(x_{n}, x_{n+1}\right)\}. \end{array} $$
Hence,
$$ d\left(x_{n}, x_{n+1}\right) \leq \psi\left(\max\left\{d\left(x_{n-1}, x_{n}\right), d\left(x_{n}, x_{n+1}\right)\right\}\right), $$
(4)
for all \(n \in \mathbb N\). Now, if there exists \(n^{*} \in \mathbb N\) such that \(p_{\alpha _{L}}(x_{n^{*}}, {Tx}_{n^{*}}) = 0\) then by Lemma 1, we have \(\phantom {\dot {i}\!}\{x_{n^{*}}\} \subset {Tx}_{n^{*}},\) that is \(x_{n^{*}} \in \, [\!{Tx}_{n^{*}}]_{\alpha _{L}}\phantom {\dot {i}\!}\) implying that \(\phantom {\dot {i}\!}x_{n^{*}}\) is an 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
$$d(x_{n}, x_{n+1}) \leq \psi (d(x_{n}, x_{n+1})) < d(x_{n}, x_{n+1}), $$
which is a contradiction. Thus, we have
$$ \begin{aligned} d\left(x_{n}, x_{n+1}\right) & \leq \psi \left(d\left(x_{n-1}, x_{n}\right)\right) \\ & \leq \psi \left(\psi\left(d\left(x_{n-2}, x_{n-1}\right)\right)\right. \\ & \vdots \\ & \leq \psi^{n} d\left(x_{0}, x_{1}\right). \end{aligned} $$
(5)
Next we show that, {x n } is a Cauchy sequence in X. Since ψΨ and continuous, then there exist ε>0 and a positive integer h=h(ε) such that
$$ \sum_{\substack{n\geq h}}{\psi^{n} d\left(x_{0}, x_{1}\right)} < \epsilon. $$
(6)
Let m>n>h. By triangular inequality, (5) and (6), we have
$$\begin{array}{*{20}l} d\left(x_{n}, x_{m}\right) & \leq \sum_{k=n}^{m-1}{d\left(x_{k}, x_{k+1}\right)} \\ & \leq \sum_{k=n}^{m-1} \psi^{k} {d\left(x_{0}, x_{1}\right)} \\ & \leq \sum_{n \geq h}{\psi^{n} d\left(x_{0}, x_{1}\right)} < \epsilon. \end{array} $$
Thus, {x n } is Cauchy sequence and since X is complete therefore we have bX 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
$$ \begin{aligned} d(b, [\!Tb]_{\alpha_{L}}) & \leq d(b, x_{n+1}) + d\left(x_{n+1}, [\!Tb]_{\alpha_{L}}\right) \\ & \leq d(b, x_{n+1}) + H\left([\!{Tx}_{n}]_{\alpha_{L}}, [\!Tb]_{\alpha_{L}}\right) \\ & \leq d(b, x_{n+1}) + D_{\alpha_{L}}\left({Tx}_{n}, Tb\right) \\ & \leq d(b, x_{n+1}) + \beta(x_{n}, b) D_{\alpha_{L}}({Tx}_{n}, Tb) \\ & \leq \psi(\Omega(x_{n}, b)) + K \min \left\{p_{\alpha_{L}}(x_{n}, {Tx}_{n}),p_{\alpha_{L}}(b, Tb), p_{\alpha_{L}}(x_{n}, Tb), p_{\alpha_{L}}(b, {Tx}_{n})\right\} \\ & \leq \psi \bigg(\max \biggl\{d(x_{n}, b), p_{\alpha_{L}}(x_{n}, {Tx}_{n}), p_{\alpha_{L}}(b, Tb), \frac{p_{\alpha_{L}}(x_{n}, Tb) + p_{\alpha_{L}}(b, {Tx}_{n})}{2} \biggr\} \bigg) \\ &\quad + K \min \left\{p_{\alpha_{L}}(x_{n}, {Tx}_{n}), p_{\alpha_{L}}(b, Tb), p_{\alpha_{L}}(x_{n}, Tb), p_{\alpha_{L}}(b, {Tx}_{n})\right\} \\ & = \psi(p_{\alpha_{L}}(b, Tb)). \end{aligned} $$
(7)
Letting n in (7), we have
$$\begin{array}{*{20}l} d\left(b, [\!Tb]_{\alpha_{L}}\right) & \leq \psi\left(p_{\alpha_{L}}(b, Tb)\right) \\ & < p_{\alpha_{L}}(b, Tb) \\ & = d\left(b, [\!Tb]_{\alpha_{L}}\right), \end{array} $$
a contraction. Hence,
$$b \in\, [\!Tb]_{\alpha_{L}}, \qquad\quad \alpha_{L}\in L\backslash \{0_{L}\}. $$

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

Example 2

Let X=[ 0,1], d(x,y)=|xy| for all x,yX, 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:
$$T(x)(t) = \left\{\begin{array}{ll} \tau, & \text{if}\,\, 0 \leq t \leq \frac{x}{6};\\ \kappa, & \text{if}\,\,\frac{x}{6} < t \leq \frac{x}{4};\\ \eta, & \text{if}\,\,\frac{x}{4} < t \leq \frac{x}{2};\\ \omega, & \text{if}\,\,\frac{x}{2} < t \leq 1. \end{array} \right. $$
For every xX, α L =τ exists for which
$$[\!Tx]_{\tau} = \left[\!0, \frac{x}{6}\right]. $$
Define β:X×X→[ 0,) as below:
$$\beta(x,y) = \left\{\begin{array}{ll} 1, & \text{if\,\(x=y\)};\\ x+1, & \text{if\,\(x \not= y\)}. \end{array} \right.$$
Then, it is easy to see that T is \(\beta _{F_{L}}\)-admissible. For each x,yX we have
$$\begin{array}{*{20}l} \beta(x,y) D_{\alpha_{L}}(Tx, Ty) & = \beta(x,y) H\left([\!Tx]_{\alpha_{L}}, [\!Ty]_{\alpha_{L}}\right) \\ & = \beta(x,y) H\bigg(\bigg[\!0, \frac{x}{6} \bigg], \bigg[\!0, \frac{y}{6} \bigg]\bigg) \\ & = \frac{1}{6} \beta(x,y) |x-y| \\ & = \frac{1}{6} \beta(x,y) d(x, y) \\ & < \frac{1}{3} d(x, y) \\ & \leq \psi (\Omega(x, y)) \\ & \quad + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}. \end{array} $$

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 0X such that 0 [ T0] τ .

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

Definition 12

Let (X,d)be a metric space, β:X×X→[ 0,) and T:XF L (X). A mapping T is said to be \(\beta _{F_{L}}^{*}\)-admissible if
$$\text{for}\ x,y \in X, \alpha_{L} \in L \backslash \{0_{L}\}, \beta(x, y) \geq 1 \Longrightarrow \beta^{*}\left([\!Tx]_{\alpha_{L}}, [\!Ty]_{\alpha_{L}}\right) \geq 1, $$
where
$$\beta^{*}\left([\!Tx]_{\alpha_{L}}, [\!Ty]_{\alpha_{L}}\right) := \inf{\left\{\beta(a,b) : a \in\, [\!Tx]_{\alpha_{L}}\ \text{and}\ b\in\, [\!Ty]_{\alpha_{L}}\right\}}. $$

Theorem 2

Let (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 ψΨ and β:X×X→[ 0,) such that for all x,yX,
$$ \begin{aligned} \beta(x, y) D_{\alpha_{L}} (Tx, Ty) \leq \psi & (\Omega(x, y)) \\ & + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, \end{aligned} $$
where K≥0 and
$$\Omega(x, y) = \max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}. $$
If the following conditions hold,
  1. 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,

     
  2. ii.

    there exist x 0X and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that β(x 0,x 1)≥1,

     
  3. iii.

    T is \(\beta _{F_{L}}^{*}\)-admissible,

     
  4. 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

Let (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 ψΨ and β:X×X→[ 0,) such that for all x,yX,
$$ \beta(x, y) D_{\alpha_{L}} (Tx, Ty) \leq \psi \bigg(\max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}\bigg). $$
If the following conditions hold,
  1. 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,

     
  2. ii.

    there exist x 0X and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that β(x 0,x 1)≥1,

     
  3. iii.

    T is \(\beta _{F_{L}}\)-admissible (or \(\beta ^{*}_{F_{L}}\)-admissible),

     
  4. iv.

    ψ is continuous.

     

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

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

Corollary 2

Let (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 ψΨ such that for all x,yX,
$$D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, $$
where K≥0 and
$$\Omega(x, y) = \max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}. $$

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

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

Corollary 3

Let (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 ψΨ such that for all x,yX,
$$ \begin{aligned} D_{\alpha_{L}} & (Tx, Ty) \\ & \leq \psi \bigg(\max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}\bigg). \end{aligned} $$

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

Remark 4

  1. i

    If we consider L= [ 0,1] in Theorem 1 and 2, Corollary 1, 2 and 3 we get Theorem 1, 2 Corollary 2, 4 and 5 of [ 21 ] respectively;

     
  2. ii

    If α L =1 L in Theorem 1 and 2, Corollary 1, 2 and 3, then by Remark 2 the L-fuzzy mappings T has atleast a 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,yX are said to be comparable if xy or yx holds.

For a partially ordered set (X,), we define
$$\barwedge := \left\{(x, y) \in X \times X : x \preceq y\ \text{or}\ y \preceq x\right\}. $$

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 xX 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

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 ψΨ such that for all \((x,y) \in \barwedge,\)
$$ D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\}, $$
(8)
where K≥0 and
$$\Omega(x, y) = \max \left\{ d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}. $$
If the following conditions hold,
  1. I.

    X satisfies the order sequential limit property,

     
  2. II.

    there exist x 0X and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that \((x_{0}, x_{1}) \in \barwedge,\)

     
  3. III.

    T is comparative L-fuzzy mapping,

     
  4. IV.

    ψ is continuous.

     

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

Proof

Let β:X×X→[ 0,) be defined as:
$$\beta(x, y) = \left\{ \begin{array}{ll} 1 & \text{if}\,\, (x, y) \in \barwedge;\\ 0 & \text{if}\,\, (x, y) \notin \barwedge. \end{array} \right. $$
Now by condition (II), we have 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,yX, we have
$$ \begin{aligned} \beta(x, y) & D_{\alpha_{L}} (Tx, Ty) \\ & \leq \psi(\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}. \end{aligned} $$
(9)

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

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 ψΨ such that for all \((x,y) \in \barwedge,\)
$$D_{\alpha_{L}} (Tx, Ty) \leq \psi \bigg(\max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\} \bigg). $$
If the following conditions hold,
  1. I.

    X satisfies the order sequential limit property,

     
  2. II.

    there exist x 0X and \(x_{1} \in \, [\!{Tx}_{0}]_{\alpha _{L}}\phantom {\dot {i}\!}\) such that \((x_{0}, x_{1}) \in \barwedge,\)

     
  3. III.

    T is comparative L-fuzzy mapping,

     
  4. 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

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 ψΨ such that for all \((x,y) \in \barwedge,\)
$$D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, $$
where K≥0 and
$$\Omega(x, y) = \max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}. $$

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

$$D_{\alpha_{L}} (Tx, Ty) \leq \psi \left(\max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}\right). $$
Then, T has at least an L-fuzzy fixed point.

Remark 5

  1. i.

    If we consider L= [ 0,1] in Theorem 3 and Corollary 4 above, we get Theorem 3 and Corollary 7 of [ 21 ], respectively;

     
  2. ii.

    If α L =1 L in Theorem 3, Corollary 4, 5 and 6, then by Remark 2 the L-fuzzy mappings T has at least a 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

(1)
Department of Mathematics, Usmanu Danfodiyo University
(2)
Department of Mathematics, COMSATS Institute of Information Technology

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