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L-fuzzy Fixed Point Theorems for L-fuzzy Mappings via \(\beta _{F_{L}}\)-admissible with Applications
- Muhammad Sirajo Abdullahi^{1}Email authorView ORCID ID profile and
- Akbar Azam^{2}
https://doi.org/10.1186/s40467-017-0056-5
© The Author(s) 2017
- Received: 9 September 2016
- Accepted: 8 February 2017
- Published: 20 February 2017
Abstract
In this paper, the authors use the idea of \(\beta _{F_{L}}\)-admissible mappings to prove some L-fuzzy fixed point theorems for a generalized contractive L-fuzzy mappings. Some examples and applications to L-fuzzy fixed points for L-fuzzy mappings in partially ordered metric spaces are also given, to support main findings.
Keywords
- L-fuzzy sets
- L-fuzzy fixed points
- L-fuzzy mappings
- \(\beta _{F_{L}}\)-admissible mappings
AMS Subject Classification
- Primary 46S40
- Secondary 47H10
- 54H25
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}.
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
- 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.
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).
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
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
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
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
- 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
Next, we give an example to support the validity of our result.
Example 2
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
Theorem 2
- 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
- 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
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
Then, T has atleast an L-fuzzy fixed point.
Applications
In this section, we establish as an application the existence of an L-fuzzy fixed point theorems in complete partially ordered metric spaces.
Below, we present some results which are essential in the remaining part of our work.
Definition 13
Let X be a nonempty set. Then, (X,d,≼) is said to be an ordered metric space if (X,d) is a metric space and (X,≼) is a partially ordered set.
Definition 14
Let (X,≼) be a partially ordered set. Then, x,y∈X are said to be comparable if x≼y or y≼x holds.
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
- 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
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
- 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
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 \((x,y) \in \barwedge,\)
Declarations
Acknowledgements
The authors thank the Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan for providing excellent research facilities.
Authors’ contributions
Both authors contributed to the writing of this paper. Both authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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