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

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.

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→2X, where 2X 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→2X, where 2X 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→2X 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.

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Acknowledgements

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

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Sirajo Abdullahi, M., Azam, A. L-fuzzy Fixed Point Theorems for L-fuzzy Mappings via \(\beta _{F_{L}}\)-admissible with Applications. J. Uncertain. Anal. Appl. 5, 2 (2017). https://doi.org/10.1186/s40467-017-0056-5

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