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Lfuzzy Fixed Point Theorems for Lfuzzy Mappings via \(\beta _{F_{L}}\)admissible with Applications
Journal of Uncertainty Analysis and Applications volume 5, Article number: 2 (2017)
Abstract
In this paper, the authors use the idea of \(\beta _{F_{L}}\)admissible mappings to prove some Lfuzzy fixed point theorems for a generalized contractive Lfuzzy mappings. Some examples and applications to Lfuzzy fixed points for Lfuzzy mappings in partially ordered metric spaces are also given, to support main findings.
Introduction
Solving realworld 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 Lfuzzy 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 1level sets of a fuzzy contraction on a complete metric space, where the 1level 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 βadmissible mapping was introduced by Samet et al. [18] for a singlevalued mappings and proved the existence of fixed point theorems via this concept, while Asl et al. [19] extended the notion to α−ψmultivalued mappings. Afterwards, Mohammadi et al. [20] established the notion of βadmissible mapping for the multivalued mappings (different from the β _{∗}admissible mapping provided in [19]).
Recently, Phiangsungnoen et al. [21] use the concept of βadmissible 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 Lfuzzy mappings and utilized it to proved a common Lfuzzy fixed point theorem. The notions of \(d_{L}^{\infty }\)metric and Hausdorff distances for Lfuzzy sets were introduced by Rashid et al. [23], they presented some fixed point theorems for Lfuzzy set valuedmappings and coincidence theorems for a crisp mapping and a sequence of Lfuzzy 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 Lfuzzy 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 Lfuzzy 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
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 αlevel 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 Lfuzzy 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 Lfuzzy set is the special case of fuzzy sets in the original sense of Zadeh [ 1 ], which shows that Lfuzzy set is larger.
Let \({\mathcal F}_{L}(X)\) denotes the class of all Lfuzzy subsets of X. Define \(\mathcal Q_{L}(X) \subset \mathcal F_{L}(X)\) as below:
The α _{ L }level set of an Lfuzzy 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
If \(A_{\alpha _{L}}, B_{\alpha _{L}}\in CB(X)\ \text {for each}\ \alpha _{L}\in L\backslash \{0_{L}\}\), 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 Lfuzzy mapping, if T is a mapping from X to \({\mathcal F}_{L}(Y)\)(i.e class of Lfuzzy subsets of Y). An Lfuzzy mapping T is an Lfuzzy 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 Lfuzzy 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 Lfuzzy 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 Lfuzzy 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
where
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 βadmissible 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 βadmissible mapping.
Example 1
Let X=[ 0,∞) and d(x,y)=x−y. Define T:X→2^{X} and β:X×X→[ 0,∞) by
and
Then, T is βadmissible.
Main Result
Lfuzzy 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 Lfuzzy 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 nondecreasing 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 βadmissible in the sense of Mohammadi et al. [20] for Lfuzzy 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 Lfuzzy fixed point theorem for some generalized type of contraction Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,
where K≥0 and
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 Lfuzzy 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
By (2) and the fact that β(x _{0},x _{1})≥1, we have
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
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
Continuing in this pattern, a sequence {x _{ n }} is obtained such that, for each n∈N, \(x_{n} \in [\!{Tx}_{n1}]_{\alpha _{L}}\phantom {\dot {i}\!}\) with β(x _{ n−1},x _{ n })≥1, we have
where
Hence,
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 Lfuzzy 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
which is a contradiction. Thus, we have
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
Let m>n>h. By triangular inequality, (5) and (6), we have
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
Letting n→∞ in (7), we have
a contraction. Hence,
□
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:
For every x∈X, α _{ L }=τ exists for which
Define β:X×X→[ 0,∞) as below:
Then, it is easy to see that 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∈ [ T0]_{ τ }.
Below, we introduce the concept of β _{∗}admissible for Lfuzzy 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
where
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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,
where K≥0 and
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 Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ and β:X×X→[ 0,∞) such that for all x,y∈X,
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 Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all x,y∈X,
where K≥0 and
Then, T has atleast an Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all x,y∈X,
Then, T has atleast an Lfuzzy fixed point.
Remark 4
Applications
In this section, we establish as an application the existence of an Lfuzzy 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
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 Lfuzzy 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 Lfuzzy fixed point theorem for Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all \((x,y) \in \barwedge,\)
where K≥0 and
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 Lfuzzy mapping,

IV.
ψ is continuous.
Then, T has atleast an Lfuzzy fixed point.
Proof
Let β:X×X→[ 0,∞) be defined as:
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 Lfuzzy 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 Lfuzzy fixed point for Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all \((x,y) \in \barwedge,\)
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 Lfuzzy mapping,

IV.
ψ is continuous.
Then, T has at least an Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all \((x,y) \in \barwedge,\)
where K≥0 and
Then, T has at least an Lfuzzy 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 Lfuzzy mapping. Suppose that there exist ψ∈Ψ such that for all \((x,y) \in \barwedge,\)
Then, T has at least an Lfuzzy fixed point.
Remark 5
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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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Both authors contributed to the writing of this paper. Both authors read and approved the final manuscript.
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The authors declare that they have no competing interests.
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Keywords
 Lfuzzy sets
 Lfuzzy fixed points
 Lfuzzy mappings
 \(\beta _{F_{L}}\)admissible mappings
AMS Subject Classification
 Primary 46S40
 Secondary 47H10
 54H25