Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces
- Mujahid Abbas^{1},
- Naeem Saleem^{2} and
- Manuel De la Sen^{3}Email author
https://doi.org/10.1186/s13663-016-0534-3
© Abbas et al. 2016
Received: 16 September 2015
Accepted: 23 March 2016
Published: 1 April 2016
Abstract
In this paper, we introduce best proximal contractions in complete ordered non-Archimedean fuzzy metric space and obtain some proximal results. The obtained results unify, extend, and generalize some comparable results in the existing literature.
Keywords
MSC
1 Introduction and preliminaries
In 1969, Fan [1], introduced the concept of a best approximation in Hausdorff locally convex topological vector spaces as follows.
Theorem 1.1
Let X be a nonempty compact convex set in a Hausdorff locally convex topological vector space E and \(T:X\rightarrow E\) a continuous mapping, then there exists a fixed point x in X, or there exist a point \(x_{0}\in X\) and a continuous semi-norm p on E satisfying \(\min_{y\in X}p(y-Tx_{0})=p(x_{0}-T(x_{0}))>0\).
A fixed point problem is to find a point x in A such that \(Tx=x\). There are certain situations where solving an equation \(d(x,Tx)=0\) for x in A is not possible, then a compromise is made on the point x in A where \(\inf\{d(y,Tx):y\in A\}\) is attained, that is, \(d(x,Tx)=\inf\{ d(y,Tx):y\in A\}\) holds. Such a point is called an approximate fixed point of T or an approximate solution of an equation \(Tx=x\). It is significant to study the conditions that ensure the existence and uniqueness of an approximate fixed point of the mapping T.
For more results in this direction, we refer to [2–7] and references therein.
On the other hand, Zadeh [8] introduced the concept of fuzzy sets. Meanwhile Kramosil and Michalek [9] defined fuzzy metric spaces. Later, George and Veeramani [10, 11] further modified the notion of fuzzy metric spaces with the help of a continuous t-norm and generalized the concept of a probabilistic metric space to the fuzzy situation. In this direction, Vetro and Salimi [12] obtained best proximity theorems in non-Archimedean fuzzy metric spaces.
The aim of this paper is to obtain a coincidence best proximity point solution of \(M(gx,Tx,t)=M(A,B,t)\) over a nonempty subset A of a partially ordered non-Archimedean fuzzy metric space X, where T is a nonself mapping and g is a self mapping on A. Our results unify, extend, and strengthen various results in [13].
Let us recall some definitions.
Definition 1.2
([14])
- (1)
∗ is associative, commutative and continuous;
- (2)
\(a\ast1=a\) for all \(a\in [0,1]\);
- (3)
\(a\ast b\leq c\ast d\) whenever \(a\leq c\) and \(b\leq d\).
Typical examples of continuous t-norm are ∧, ⋅, and \(\ast _{L}\), where, for all \(a,b\in[0,1]\), \(a\wedge b=\min\{a,b\} \), \(a\cdot b=ab\), and \(\ast_{L}\) is the Lukasiewicz t-norm defined by \(a\ast _{L}b=\max\{a+b-1,0\}\).
It is easy to check that \(\ast_{L}\leq\cdot\leq\wedge\). In fact ∗ ≤ ∧ for all continuous t-norms ∗.
Definition 1.3
([11])
- (i)
\(M(x,y,t)>0\),
- (ii)
\(x=y\) if and only if \(M(x,y,t)=1\) for all \(t>0\),
- (iii)
\(M(x,y,t)=M(y,x,t)\),
- (iv)
\(M(x,z,t+s)\geq M(x,y,t)\ast M(y,z,s)\) for all \(t,s>0\),
- (v)
\(M(x,y,\cdot):[0,\infty)\rightarrow[0,1]\) is left continuous.
The triplet \((X,M,\ast)\) is called a fuzzy metric space.
Since M is a fuzzy set on \(X\times X\times[0,\infty)\), the value \(M(x,y,t)\) is regarded as the degree of closeness of x and y with respect to t.
It is well known that for each \(x,y\in X\), \(M(x,y,\cdot)\) is a nondecreasing function on \((0,+\infty)\) [15].
- (vi)
\(M(x,z,\max\{t,s\})\geq M(x,y,t)\ast M(y,z,s)\) for all \(t,s>0\),
As (vi) implies (iv), every non-Archimedean fuzzy metric space is a fuzzy metric space. Also, if we take \(s=t\), then (vi) reduces to \(M(x,z,t)\geq M(x,y,t)\ast M(y,z,t)\) for all \(t>0\). And M in this case is said to be a strong fuzzy metric on X.
Note that a sequence \(\{x_{n}\}\) converges to \(x\in X\) (with respect to \(\tau_{M}\)) if and only if \(\lim_{n\rightarrow\infty }M(x_{n},x,t)=1 \) for all \(t>0\).
A sequence \(\{x_{n}\}\) in a fuzzy metric space X is said to be a Cauchy sequence if for each \(t>0\) and \(\varepsilon\in(0,1)\), there exists \(n_{0}\in\mathbb{N}\) such that \(M(x_{n},x_{m},t)>1-\varepsilon\) for all \(n,m\geq n_{0}\). A fuzzy metric space X is complete [11] if every Cauchy sequence converges in X. A subset A of X is closed if for each convergent sequence \(\{x_{n}\}\) in A with \(x_{n}\longrightarrow x\), we have \(x\in A\). A subset A of X is compact if each sequence in A has a convergent subsequence.
Definition 1.5
([7])
Definition 1.6
([4])
- (i)
ψ is continuous and nondecreasing on \((0,1)\) and \(\psi (t)>t\) also \(\psi(0)=0\) and \(\psi(1)=1\).
- (ii)
\(\lim_{n\rightarrow\infty}\psi^{n}(t)=1\) if and only if \(t=1\).
- (i)
η is continuous and strictly decreasing on \((0,1)\) and \(\eta(t)< t\) for all \(t\in(0,1)\),
- (ii)
\(\eta(1)=1\) and \(\eta(0)=0\).
If we take \(\eta(t)=2t-t^{2}\), then \(\eta\in\Lambda\) and hence \(\Lambda \neq\phi\).
2 Best proximity point in partially ordered non-Archimedean fuzzy metric space
Definition 2.1
Let A be a nonempty subset of a non-Archimedean fuzzy metric space \((X,M,\ast)\). A self mapping f on A is said to be (a) fuzzy isometry if \(M(fx,fy,t)=M(x,y,t)\) for all \(x,y\in A \) and \(t>0\) (b) fuzzy expansive if, for any \(x,y\in A \) and \(t>0\), we have \(M(fx,fy,t)\leq M(x,y,t)\), (c) fuzzy nonexpansive if, for any \(x,y\in A \) and \(t>0\), we have \(M(fx,fy,t)\geq M(x,y,t)\).
Example 2.2
Let \(X=[0,1]\times \mathbb{R} \) and \(d:X\times X\rightarrow \mathbb{R} \) be a usual metric on X. Let \(A=\{(0,x):x\in \mathbb{R} \}\). Note that \((X,M_{d},\cdot)\) is non-Archimedean fuzzy metric space, where \(M_{d}\) is standard fuzzy metric induced by d. Define the mapping \(f:A\rightarrow A\) by \(f(0,x)=(0,-x)\). Note that \(M_{d}(w,u,t)=\frac{t}{ t+\vert x-y\vert }=M(fw,fu,t)\), where \(w=(0,x)\), \(u=(0,y)\in A\).
Note that every fuzzy isometry is fuzzy expansive but the converse does not hold in general.
Example 2.3
Example 2.4
Note that the fuzzy expansive and nonexpansive mapping are fuzzy isometries. However, the converse is not true in general.
Definition 2.5
Let A, B be nonempty subsets of a non-Archimedean fuzzy metric space \((X,M,\ast)\). A set B is said to be fuzzy approximatively compact with respect to A if for every sequence \(\{y_{n}\}\) in B and for some \(x\in A\), \(M(x,y_{n},t)\longrightarrow M(x,B,t)\) implies that \(x\in A_{0}(t)\).
Definition 2.6
([17])
A sequence \(\{t_{n}\}\) of positive real numbers is said to be s-increasing if there exists \(n_{0}\in \mathbb{N} \) such that \(t_{n+1}\geq t_{n}+1 \) for all \(n\geq n_{0}\).
Definition 2.7
(compare [18])
A fuzzy metric space \((X,M,\ast) \) is said to satisfy property T if, for any s-increasing sequence, there exists \(n_{0}\in \mathbb{N} \) such that \(\prod_{n\geq n_{0}}^{\infty}M(x,y,t_{n})\geq 1-\varepsilon \) for all \(n\geq n_{0}\).
A 4-tuple \((X,M,\ast,\preceq)\) is called a partially ordered fuzzy metric space if \((X,\preceq)\) is a partially ordered set and \((X,M,\ast)\) is a non-Archimedean fuzzy metric space. Unless otherwise stated, it is assumed that A, B are nonempty closed subsets of partially ordered fuzzy metric space \((X,M,\ast,\preceq)\).
Definition 2.8
([13])
A mapping \(T:A\longrightarrow B\) is called (a) nondecreasing or order preserving if, for any x, y in A with \(x\preceq y\), we have \(Tx\preceq Ty\); (b) an ordered reversing if, for any x, y in A with \(x\preceq y\), we have \(Tx\succeq Ty\); (c) monotone if it is order preserving or order reversing.
Definition 2.9
([19])
Let A, B be nonempty subsets of partially ordered fuzzy metric space \((X,M,\ast,\preceq)\) and \(\psi :[0,1]\longrightarrow[0,1]\) be a continuous mapping. A mapping \(T:A\longrightarrow B\) is said to be a fuzzy ordered ψ-contraction if, for any \(x,y\in A\) with \(x\preceq y\), we have \(M(Tx,Ty,t)\geq\psi [M(x,y,t)]\) for all \(t>0\).
Definition 2.10
Definition 2.11
Definition 2.12
Definition 2.13
If \(A=B\), then a proximal fuzzy order preserving mapping will become fuzzy order preserving.
Definition 2.14
If \(A=B\), then proximal fuzzy order reversing mapping will become fuzzy order reversing.
Definition 2.15
From now on, we use the notation \(\Delta_{(t)}\) for a set \(\{(x,y)\in A_{0}(t)\times A_{0}(t): \mbox{either } x\preceq y\mbox{ or }{y\preceq x} \}\).
We start with the following result.
Theorem 2.16
Proof
Example 2.17
The above example shows that our result is a potential generalization of Theorem 3.1 in [13].
Corollary 2.18
Proof
Every fuzzy isometry is fuzzy expansive, and this corollary satisfies all the conditions of Theorem 2.16. □
Example 2.19
Corollary 2.20
Proof
This corollary satisfies all the conditions of Theorem 2.16 by taking \(gx=I_{A}\) (an identity mapping on A). □
3 Best proximity point in partially ordered non-Archimedean fuzzy metric spaces for proximal ψ-contractions of type-II
Theorem 3.1
Proof
Example 3.2
Corollary 3.3
Proof
Here the T satisfy all the conditions of Theorem 3.1 if we consider g as fuzzy isometry mapping. □
Corollary 3.4
Let \(T:A\rightarrow B\) is continuous, proximally monotone, and proximal ordered fuzzy ψ-contraction of type-II. Suppose that each pair of elements in X has a lower and upper bound, and an s-increasing sequence \(\{t_{n}\}\) satisfying property T, for any \(t>0\), \(A_{0}(t)\) and \(B_{0}(t)\) are nonempty such that \(T(A_{0}(t))\subseteq B_{0}(t)\).
Then there exists a unique element \(x^{\ast}\in A\) such that \(M(x^{\ast },Tx^{\ast},t)=M(A,B,t)\). Further, for any fixed element \(x_{0}\in A_{0}(t)\), the sequence \(\{x_{n}\}\in A_{0}(t)\), defined by \(M(x_{n+1},Tx_{n},t)=M(A,B,t)\), converges to \(x^{\ast}\).
Proof
Here the T satisfy all the conditions of Theorem 3.1 if \(g(x)=I_{A}\) (an identity mapping on A). □
4 Best proximity point in partially ordered non-Archimedean fuzzy metric spaces for proximal η-contractions
Theorem 4.1
Let \(T:A\rightarrow B\) be continuous, proximally monotone, and proximal fuzzy ordered η-contraction such that, for any \(t>0\), \(A_{0}(t) \) and \(B_{0}(t)\) are nonempty with \(T(A_{0}(t))\subseteq B_{0}(t)\), \(g:A\rightarrow A\) surjective, fuzzy nonexpansive and inverse monotone mapping with \(A_{0}(t)\subseteq g(A_{0}(t))\) for any \(t>0\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that \(M(gx_{1},Tx_{0},t)=M(A,B,t)\) with \((x_{0},x_{1})\in\Delta_{(t)}\), then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast },Tx^{\ast},t)=M(A,B,t)\) provided that each pair of elements in X has a lower and upper bound. Further, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\} \) defined by \(M(g \overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\) converges to \(x^{\ast}\).
Proof
Example 4.2
Declarations
Acknowledgements
M De la Sen thanks the Spanish Ministry of Economy and Competitiveness for partial support of this work through Grant DPI2012-30651. He also thanks the Basque Government for its support through Grant IT378-10, and the University of Basque Country for its support through Grant UFI 11/07.
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.
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