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 [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}.

Let A,B∈C 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 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

A partially ordered set (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.

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 $A,B\in {\mathcal F}_{L}(X)$, 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

$$D_{\alpha_{L}}(A,B) = H(A_{\alpha_{L}}, B_{\alpha_{L}}). $$

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 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:

$$\chi_{L_{A}} = \left\{\begin{array}{ll} 0_{L} & \text{if}\ x \notin A;\\ 1_{L} & \text{if}\ x \in A. \end{array} \right.$$

Definition 7

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

(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 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

Let X=[ 0,∞) and d(x,y)=|x−y|. 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,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

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 ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,

$$ \begin{aligned} \beta(x, y) D_{\alpha_{L}} (Tx, Ty) &\leq \psi (\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, \end{aligned} $$

(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,

• 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

For 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

$$ d(x_{1},x_{2}) = p_{\alpha_{L}}(x_{1}, {Tx}_{1}) \leq D_{\alpha_{L}}({Tx}_{0},{Tx}_{1}). $$

(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 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

$$ d(x_{2},x_{3}) = p_{\alpha_{L}}(x_{2}, {Tx}_{2}) \leq D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}). $$

(3)

For 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

$$\begin{array}{*{20}l} d(x_{2},x_{3}) & \leq D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}) \\ & \leq \beta(x_{1}, x_{2}) D_{\alpha_{L}}({Tx}_{1},{Tx}_{2}) \\ & \leq \psi(\Omega(x_{1}, x_{2})) + K \min \left\{p_{\alpha_{L}}(x_{1}, {Tx}_{1}), p_{\alpha_{L}}(x_{2}, {Tx}_{2}),\right. \\ &\quad \left. p_{\alpha_{L}}(x_{1}, {Tx}_{2}), p_{\alpha_{L}}(x_{2}, {Tx}_{1})\right\} \\ & \leq \psi(\Omega(x_{1}, x_{2})) + K \min \left\{p_{\alpha_{L}}(x_{1}, x_{2}), p_{\alpha_{L}}(x_{2}, x_{3}), p_{\alpha_{L}}(x_{1}, x_{3}), 0\right\} \\ & = \psi(\Omega(x_{1}, x_{2})). \end{array} $$

Continuing in this pattern, a sequence {x _n } is obtained such that, for each n∈N, $x_{n} \in [\!{Tx}_{n-1}]_{\alpha _{L}}\phantom {\dot {i}\!}$ 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 ψ∈Ψ 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 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

$$ \begin{aligned} d(b, [\!Tb]_{\alpha_{L}}) & \leq d(b, x_{n+1}) + d\left(x_{n+1}, [\!Tb]_{\alpha_{L}}\right) \\ & \leq d(b, x_{n+1}) + H\left([\!{Tx}_{n}]_{\alpha_{L}}, [\!Tb]_{\alpha_{L}}\right) \\ & \leq d(b, x_{n+1}) + D_{\alpha_{L}}\left({Tx}_{n}, Tb\right) \\ & \leq d(b, x_{n+1}) + \beta(x_{n}, b) D_{\alpha_{L}}({Tx}_{n}, Tb) \\ & \leq \psi(\Omega(x_{n}, b)) + K \min \left\{p_{\alpha_{L}}(x_{n}, {Tx}_{n}),p_{\alpha_{L}}(b, Tb), p_{\alpha_{L}}(x_{n}, Tb), p_{\alpha_{L}}(b, {Tx}_{n})\right\} \\ & \leq \psi \bigg(\max \biggl\{d(x_{n}, b), p_{\alpha_{L}}(x_{n}, {Tx}_{n}), p_{\alpha_{L}}(b, Tb), \frac{p_{\alpha_{L}}(x_{n}, Tb) + p_{\alpha_{L}}(b, {Tx}_{n})}{2} \biggr\} \bigg) \\ &\quad + K \min \left\{p_{\alpha_{L}}(x_{n}, {Tx}_{n}), p_{\alpha_{L}}(b, Tb), p_{\alpha_{L}}(x_{n}, Tb), p_{\alpha_{L}}(b, {Tx}_{n})\right\} \\ & = \psi(p_{\alpha_{L}}(b, Tb)). \end{aligned} $$

(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)=|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:

$$T(x)(t) = \left\{\begin{array}{ll} \tau, & \text{if}\,\, 0 \leq t \leq \frac{x}{6};\\ \kappa, & \text{if}\,\,\frac{x}{6} < t \leq \frac{x}{4};\\ \eta, & \text{if}\,\,\frac{x}{4} < t \leq \frac{x}{2};\\ \omega, & \text{if}\,\,\frac{x}{2} < t \leq 1. \end{array} \right. $$

For every x∈X, 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,y∈X 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 0∈X 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:X→F _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 ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,

$$ \begin{aligned} \beta(x, y) D_{\alpha_{L}} (Tx, Ty) \leq \psi & (\Omega(x, y)) \\ & + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, \end{aligned} $$

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,

• 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

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 ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,

$$ \beta(x, y) D_{\alpha_{L}} (Tx, Ty) \leq \psi \bigg(\max \left\{d(x, y), p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), \frac{p_{\alpha_{L}}(x, Ty) + p_{\alpha_{L}}(y, Tx)}{2}\right\}\bigg). $$

If the following conditions hold,

• 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

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 ψ∈Ψ such that for all x,y∈X,

$$D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, $$

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,y∈X 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 ψ∈Ψ such that for all x,y∈X,

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

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

Remark 4

• 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;

• 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,y∈X are said to be comparable if x≼y or y≼x 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 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

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 $(x,y) \in \barwedge,$

$$ D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\}, $$

(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,

• 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

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,y∈X, 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 ψ∈Ψ 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,

• 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

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 $(x,y) \in \barwedge,$

$$D_{\alpha_{L}} (Tx, Ty) \leq \psi(\Omega(x, y)) + K \min \left\{p_{\alpha_{L}}(x, Tx), p_{\alpha_{L}}(y, Ty), p_{\alpha_{L}}(x, Ty), p_{\alpha_{L}}(y, Tx)\right\}, $$

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 ψ∈Ψ 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

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

• 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.