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.
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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.
Let A and B be two nonempty subsets of X and \(T:A\rightarrow B\). Suppose that \(d(A,B):=\inf\{d(x,y):x\in A \mbox{ and } y\in B\}\) is the distance between two sets A and B where \(A\cap B=\phi\). A point \(x^{\ast }\) is called a best proximity point of T if \(d(x^{\ast},Tx^{\ast})=d(A,B)\). Indeed, if T is a multifunction from A to B then
for all \(x\in A\), always. Note that if \(A=B\), then the best proximity point will reduce to a fixed point of the mapping T. Hence the results dealing with the best proximity point problem extend fixed point theory in a natural way.
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])
A binary operation \(\ast:[0,1]^{2}\longrightarrow [0,1]\) is called a continuous t-norm if
-
(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])
Let X be a nonempty set, and ∗ be a continuous t-norm. A fuzzy set M on \(X\times X\times[ 0,+\infty)\) is said to be a fuzzy metric if, for any \(x,y,z\in X\), the following conditions hold:
-
(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].
If we replace (iv) with
-
(vi)
\(M(x,z,\max\{t,s\})\geq M(x,y,t)\ast M(y,z,s)\) for all \(t,s>0\),
then the triplet \((X,M,\ast)\) is said to be a non-Archimedean fuzzy metric space.
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.
Each fuzzy metric M on X generates a Hausdorff topology \(\tau_{M}\) whose base is the family of open M-balls \(\{B_{M}(x,\varepsilon,t):x\in X,\varepsilon\in(0,1),t>0\}\), where
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 \(t>0\).
Let \((X,d)\) be a metric space. Define \(M_{d}:X\times X\times[ 0,\infty)\rightarrow[0,1]\) by
Then \((X,M_{d},\cdot)\) is a fuzzy metric space and is called the standard fuzzy metric space induced by a metric d [10]. The topologies \(\tau_{M_{d}}\) and \(\tau_{d}\) (the topology induced by the metric d) on X are the same. Note that if d is a metric on a set X, then the fuzzy metric space \((X,M_{d},\ast)\) is strong for every continuous t-norm ‘∗’ such that for all ∗ ≤ ⋅, where \(M_{d}\) is the standard fuzzy metric (see [16]).
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.
Lemma 1.4
([15])
M is a continuous function on \(X^{2}\times (0,\infty)\).
Definition 1.5
([7])
Let A and B be two nonempty subsets of a fuzzy metric space \((X,M,\ast)\). We define \(A_{0}(t)\) and \(B_{0}(t)\) as follows:
The distance of a point \(x\in X\) from a nonempty set A for \(t>0\) is defined as
and the distance between two nonempty sets A and B for \(t>0\) is defined as
Definition 1.6
([4])
Let Ψ be the set of all mappings \(\psi:[0,1]\rightarrow [0,1]\) satisfying the following properties:
-
(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 \(t=1\).
Let Λ be the set of all mappings \(\eta:[0,1]\rightarrow [ 0,1]\) which satisfy the following properties:
-
(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
Let \(X=[0,4]\times \mathbb{R} \) 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 a non-Archimedean fuzzy metric space, where \(M_{d}\) is the standard fuzzy metric induced by d. Define the mapping \(f:A\rightarrow A\) by
If \(x=(0,0)\) and \(y=(0,4) \) then \(M(x,y,t)=\frac{t}{t+4} \) and \(M(fx,fy,t)=\frac{t}{t+400}\). This shows that f is fuzzy expansive but not a fuzzy isometry.
Example 2.4
Let \(X=[0,1]\times \mathbb{R} \), \(d:X\times X\rightarrow \mathbb{R} \) a usual metric on X and \(A=\{(0,x):x\in \mathbb{R} \}\). Define a mapping \(f:A\rightarrow A\) by
If \(x=(0,0)\) and \(y=(0,1) \) then \(M(x,y,t)=\frac{t}{t+1} \) and \(M(fx,fy,t)=\frac{t}{t+\frac{1}{10}}\geq\frac{t}{t+1}=M(x,y,t)\). Thus f is fuzzy nonexpansive but not a fuzzy isometry.
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
A mapping \(T:A\longrightarrow B\) is called a fuzzy ordered proximal ψ-contraction of type-I if, for any u, v, x, and y in A, the following condition holds:
Definition 2.11
A mapping \(T:A\longrightarrow B\) is said to be a fuzzy ordered proximal ψ-contraction of type-II if, for any u, v, x, and y in A, and for some \(\alpha\in(0,1)\), the following condition holds:
Definition 2.12
A mapping \(T:A\longrightarrow B\) is called a fuzzy ordered η-proximal contraction if, for any u, v, x, and y in A, the following condition holds:
Definition 2.13
A mapping \(T:A\longrightarrow B\) is said to be a proximal fuzzy order preserving if, for any u, v, x, and y in A, the following implication holds:
If \(A=B\), then a proximal fuzzy order preserving mapping will become fuzzy order preserving.
Definition 2.14
A mapping \(T:A\longrightarrow B\) is said to be a proximal fuzzy order reversing if for any u, v, x, and y in A, the following implication holds:
If \(A=B\), then proximal fuzzy order reversing mapping will become fuzzy order reversing.
Definition 2.15
A point x in A is said to be an optimal coincidence point of the pair of mappings \((g,T)\), where \(T:A\longrightarrow B\) is a nonself mapping and \(g:A\longrightarrow A\) is a self mapping if
holds.
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
Let \(T:A\rightarrow B\) be continuous, proximally monotone, and proximal fuzzy ordered ψ-contraction of type-I, \(g:A\rightarrow A\) surjective, fuzzy expansive and inverse monotone mapping. Suppose that each pair of elements in X has a lower and upper bound and for any \(t>0\), \(A_{0}(t)\) and \(B_{0}(t)\) are nonempty such that \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that
then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\), that is, \(x^{\ast}\) is an optimal coincidence point of the pair \((g,T)\). 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
Let \(x_{0}\) and \(x_{1}\) be given points in \(A_{0}(t)\) such that
Since \(Tx_{1}\in T(A_{0}(t))\subseteq B_{0}(t)\), and \(A_{0}(t)\subseteq g(A_{0}(t))\), we can choose an element \(x_{2}\in A_{0}(t)\) such that
As T is proximally monotone, we have \((gx_{1},gx_{2})\in\Delta_{(t)}\) which further implies that \((x_{1},x_{2})\in\Delta_{(t)}\). Continuing this way, we obtain a sequence \(\{x_{n}\}\) in \(A_{0}(t)\), such that it satisfies
for each positive integer n. Having chosen \(x_{n}\), one can find a point \(x_{n+1}\) in \(A_{0}(t)\) such that
Since \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\), T is proximally monotone mapping, so from (3) and (4) it follows that \((gx_{n},gx_{n+1})\in\Delta_{(t)}\) and \((x_{n},x_{n+1})\in \Delta_{(t)}\). Note that
Denote \(M(x_{n},x_{n+1},t)=\tau_{n}(t)\) for all \(t>0\), \(n\in \mathbb{N} \cup\{0\}\). The above inequality becomes
and
Thus \(\{\tau_{n}(t)\}\) is an increasing sequence for all \(t>0\). Consequently, there exists \(\tau(t)\leq1\) such that \(\lim_{n\rightarrow +\infty}\tau_{n}(t)=\tau(t)\). Note that \(\tau(t)=1\). If not, there exists some \(t_{0}>0\) such that \(\tau(t_{0})<1\). Also, \(\tau _{n}(t_{0})\leq\tau(t_{0})\). By taking limit as \(n\rightarrow\infty \) on both sides of (6), we have
a contradiction. Hence \(\tau(t)=1\). Now we show that \(\{x_{n}\}\) is a Cauchy sequence. Suppose on the contrary that \(\{x_{n}\}\) is not a Cauchy sequence, then there exist \(\varepsilon\in(0,1)\) and \(t_{0}>0\) such that for all \(k\in \mathbb{N} \), there are \(m_{k},n_{k}\in \mathbb{N} \), with \(m_{k}>n_{k}\geq k\) such that
Assume that \(m_{k}\) is the least integer exceeding \(n_{k}\) and satisfying the above inequality, then we have
So, for all k,
On taking the limit as \(k\rightarrow\infty\) on both sides of the above inequality, we obtain \(\lim_{k\rightarrow+\infty }M(x_{m_{k}},x_{n_{k}},t_{0})=1-\varepsilon\). Note that
and
imply that
From (4), we have
Thus
On taking the limit as \(k\rightarrow\infty\) in the above inequality, we get \(1-\varepsilon\geq\psi(1-\varepsilon)>1-\varepsilon\), a contradiction. Hence \(\{x_{n}\}\) is a Cauchy sequence in the closed subset \(A(t)\) of complete partially ordered fuzzy metric space \((X,M,\ast,\preceq)\). There exists \(x^{\ast}\in A(t)\) such that \(\lim_{n\rightarrow\infty }M(x_{n},x^{\ast},t)=1\), for all \(t>0\). This further implies that
Hence \(x^{\ast}\in A_{0}(t)\) is the optimal coincidence point of a pair \(\{g,T\}\). To prove the uniqueness of \(x^{\ast}\); We show that, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\}\in A_{0}(t) \) defined by \(M(g\overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\) converges to \(x^{\ast}\). Suppose that there is another element \(\overline{x}_{0} \in A(t)\) such that \(0< M(x_{0},\overline{x}_{0},t)<1\) for all \(t>0\) satisfying
Suppose that \((\overline{x}_{0},x_{0})\in \Delta_{(t)}\), that is, \(\overline{x}_{0}\preceq x_{0}\) or \(\overline{x}_{0}\succeq x_{0}\). Then by the given assumption, we have
a contradiction. So \(x^{\ast}\) is unique. If \((\overline{x}_{0},x_{0})\notin \Delta_{(t)}\), then by assumption, suppose that \(u_{0}\) be a lower bound of \(x_{0} \) and \(\overline{x}_{0}\), also assume that \(\overline{u}_{0} \) is an upper bound of \(x_{0}\) and \(\overline{x}_{0}\). That is,
Recursively, construct the sequences \(\{u_{n}\}\) and \(\{\overline {u}_{n}\}\), such that
The proximal monotonicity of the mapping T and the monotonicity of the inverse of g imply that
Since \((x_{0},u_{0})\in\Delta_{(t)}\), also \((x_{0},\overline {u}_{0})\in \Delta_{(t)}\), similarly we have \((x_{n},u_{n})\in\Delta_{(t)} \) and \((x_{n},\overline{u}_{n})\in\Delta_{(t)}\), therefore
Hence
This completes the proof. □
Example 2.17
Let \(X=[0,1]\times \mathbb{R} \) and ⪯ be the usual order on \(\mathbb{R} ^{2}\), that is, \((x,y)\preceq(z,w)\) if and only if \(x\leq z\) and \(y\leq w\). Suppose that \(A=\{(-1,x): \mbox{for all }x\in \mathbb{R} \}\) and \(B=\{(1,y):\mbox{for all } y\in \mathbb{R} \}\). \((X,M,\ast,\preceq)\) is a complete ordered metric space under \(M(x,y,t)=\frac{t}{t+d(x,y)} \) for all \(t>0\), where \(d(x,y)=\vert x_{1}-y_{1}\vert +\vert x_{2}-y_{2}\vert \) for all \(x=(x_{1},y_{1})\), \(y=(x_{2},y_{2})\). Note that \(M(A,B,t)=\frac {t}{t+2}\), \(A_{0}(t)=A\), and \(B_{0}(t)=B\). Define \(T:A\rightarrow B\) by
Let \(g:A\rightarrow A\) be defined by \(g(-1,x)=(-1,2x)\). Note that g is fuzzy expansive and its inverse is monotone. Obviously, \(T(A_{0}(t))=B_{0}(t)\), and \(A_{0}(t)=g(A_{0}(t))\). Note that \(u=(-1,\frac{y_{1}}{4})\), \(v=(-1,\frac{y_{2}}{4})\), \(x=(-1,y_{1})\), and \(y=(-1,y_{2})\in A\) satisfy
Also, note that
where \(\psi(t)=\sqrt{t}\). Thus all conditions of Theorem 2.16 are satisfied. However, \((-1,0)\) is the optimal coincidence point of g and T, satisfying the conclusion of the theorem.
The above example shows that our result is a potential generalization of Theorem 3.1 in [13].
Corollary 2.18
Let \(T:A\rightarrow B\) is continuous, proximally monotone, and proximal fuzzy ordered ψ-contraction of type-I, \(g:A\rightarrow A\) surjective, a fuzzy isometry, and an inverse monotone mapping. Suppose that each pair of elements in X has a lower and upper bound, for any \(t>0\), \(A_{0}(t)\) and \(B_{0}(t)\) are nonempty such that \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that
then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\). 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
Every fuzzy isometry is fuzzy expansive, and this corollary satisfies all the conditions of Theorem 2.16. □
Example 2.19
Let \(X=[-1,1]\times \mathbb{R} \) and ⪯ a usual order on \(\mathbb{R} ^{2}\). Let \(A=\{(-1,x): \mbox{for all }x\in \mathbb{R} \}\), \(B=\{(1,y): \mbox{for all } y\in \mathbb{R} \}\), and \((X,M,\ast,\preceq)\) a complete fuzzy ordered metric space as given in Example 2.17. Note that \(M(A,B,t)=\frac{t}{t+2}\), \(A_{0}(t)=A\) and \(B_{0}(t)=B\). Define \(T:A\rightarrow B\) by
Let \(g:A\rightarrow A\) be defined by \(g(-1,x)=(-1,-x)\). Note that g is a fuzzy isometry and its inverse is monotone. Obviously, \(T(A_{0}(t))=B_{0}(t)\), and \(A_{0}(t)=g(A_{0}(t))\). Note that \(u=(-1,-\frac{y_{1}}{5})\), \(v=(-1,-\frac{y_{2}}{5})\), \(x=(-1,y_{1})\), and \(y=(-1,y_{2})\in A_{0}(t)\) satisfy
Also, note that
where \(\psi(t)=\sqrt{t}\). All conditions of Corollary 2.18 are satisfied. Moreover, \((-1,0)\) is an optimal coincidence point of g and T.
Corollary 2.20
Let \(T:A\rightarrow B\) be a continuous, proximally monotone, and proximal fuzzy ordered ψ-contraction of type-I. Suppose that each pair of elements in X has a lower and upper bound for any \(t>0\), \(A_{0}(t)\) and \(B_{0}(t)\) are nonempty such that \(T(A_{0}(t))\subseteq B_{0}(t)\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that
then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(x^{\ast},Tx^{\ast},t)=M(A,B,t)\). Further, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\} \) defined by \(M(\overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\) converges to \(x^{\ast}\).
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
Let \(T:A\rightarrow B\) is continuous, proximally monotone, and proximal ordered fuzzy ψ-contraction of type-II, \(g:A\rightarrow A \) surjective, fuzzy expansive, and inverse monotone mapping. 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 \(T(A_{0}(t))\subseteq B_{0}(t) \) and \(A_{0}(t)\subseteq g(A_{0}(t))\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that
then there exists a unique element \(x\in A_{0}(t)\) such that \(M(gx,Tx,t)=M(A,B,t)\). Further, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\}\in A_{0}(t)\), defined by \(M(g\overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\), converges to x.
Proof
Let \(x_{0}\) and \(x_{1} \) be given elements in \(A_{0}(t)\). such that
Since \(Tx_{1}\in T(A_{0}(t))\subseteq B_{0}(t)\), \(A_{0}(t)\subseteq T(A_{0}(t))\subseteq B_{0}(t)\), and \(A_{0}(t)\subseteq g(A_{0}(t))\), it follows that there exists an element \(x_{2}\in A_{0}(t)\) such that it satisfies
As T is proximal monotone, we have \((gx_{1},gx_{2})\in\Delta_{(t)}\), which further implies that \((x_{1},x_{2}) \in\Delta_{(t)}\). Continuing this way, we obtain a sequence \(\{x_{n}\} \) in \(A_{0}(t) \) such that
for each positive integer n. Hence after finding \(x_{n}\), we can find an element \(x_{n+1}\) in \(A_{0}(t)\) such that
Since \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\), T is proximally monotone mapping, so from (15) and (16), it follows that \((gx_{n},gx_{n+1}) \in\Delta_{(t)}\) and \((x_{n},x_{n+1}) \in\Delta_{(t)}\). Note that
for all \(n\geq0\). Recursively,
for all \(t>0\) and \(m,n\in \mathbb{N} \), where \(m\geq n\), so we have
where \(t_{i}=\frac{t}{\alpha^{i}}\). As \(\lim_{n\rightarrow\infty }(t_{n+1}-t_{n})=\infty\), \(\{t_{n}\}\) is an s-increasing sequence satisfying the property T. Consequently for each \(\varepsilon>0\), there exists \(n_{0}\in \mathbb{N} \), so we have \(\prod_{n=1}^{\infty}M(x_{0},x_{1}, t_{n})\geq 1-\varepsilon\) for all \(n\geq n_{0}\). Hence we obtain \(M(x_{n},x_{m},t)\geq 1-\varepsilon\) for all \(n,m\geq n_{0}\) and \(\{x_{n}\}\) is a Cauchy sequence in \(A(t)\). By the completeness of X, there exists x in \(A(t)\) such that \(\lim_{n\rightarrow\infty}M(x_{n},x,t)=1\) for all \(t>0\). This further implies that
Since g is continuous, the sequence \(\{gx_{n}\}\) converges to gx. Therefore, \(M(gx,Tx_{n},t)\rightarrow M(gx,B,t)\). Since \(B(t)\) is fuzzy approximately compact with respect to \(A(t)\), \(\{Tx_{n}\}\) has a subsequence which converges to y in \(B(t)\) such that
for some \(y\in B(t)\), hence \(gx \in A_{0}(t)\) implies \(gx=gu \) for some \(u\in A_{0}(t)\). Hence \(M(x,u,t)\geq M(gx,gu,t)=1\), which implies that \(M(x,u,t)=1\). Thus x and u are identical, and hence \(x\in A_{0}(t)\). Since \(T(A_{0}(t))\subseteq B_{0}(t)\),
for some z in \(A(t)\). From (16) and (19) we obtain
Taking the limit as \(n\rightarrow\infty\), the above inequality becomes
which shows that \(\{gx_{n}\}\) converges to z
Since g is continuous, the sequence \(\{gx_{n}\}\) converges to gx such that
Hence we have \(gx=z\),
Suppose that there is another element \(x^{\ast}\) such that
First suppose that \((x,x^{\ast})\in\Delta_{(t)}\). From (23) and (24), it follows that
which further implies that
a contradiction. Hence x is unique.
Now, suppose that \((x,x^{\ast})\notin\Delta_{(t)}\). Let \(\overline {x}_{0} \) be any element in \(A_{0}(t)\), \(u_{0}\) and \(\overline{u}_{0}\) be lower and upper bounds of \(x_{0}\) and \(\overline{x}_{0}\), respectively such that
Recursively, we can find sequences \(\{u_{n}\}\) and \(\{\overline{u}_{n}\}\) such that
The proximal monotonicity of the mapping T and the monotonicity of the inverse of g implies that
Since \((x_{0},u_{0})\in\Delta_{(t)}\), also \((x_{0},\overline {u}_{0})\in \Delta_{(t)}\). It follows that
Hence
This completes the proof. □
Example 3.2
Let \(X=[0,2]\times \mathbb{R} \) and ⪯ a usual order on \(\mathbb{R} ^{2}\). Let \(A=\{(0,x):x\geq0\mbox{ and }x\in \mathbb{R} \}\), \(B=\{(2,y): \mbox{for all }y\in \mathbb{R} \}\), and \((X,M,\ast,\preceq)\) a complete fuzzy ordered metric space as given in Example 2.17. Note that \(A_{0}(t)=A\), \(B_{0}(t)=\{ (2,y):y\geq0 \mbox{ and }y\in \mathbb{R} \}\). Define \(T:A\rightarrow B\) by
Let \(g:A\rightarrow A\) be defined by \(g(0,x)=(0,10x)\). Note that g is a fuzzy expansive and its inverse is monotone. Obviously, \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\). Note that \(u=(0,\frac{y_{1}}{100})\), \(v=(0,\frac{y_{2}}{100})\), \(x=(0,y_{1})\), and \(y=(0,y_{2})\in A_{0}(t)\) satisfy
Also, note that
where \(\psi(t)=\sqrt{t}\) and for all \(\alpha\in[\frac{1}{10},1]\). All conditions of Theorem 3.1 are satisfied. Moreover, \((0,0)\) is optimal coincidence point of g and T.
Corollary 3.3
Let \(T:A\rightarrow B\) is continuous, proximally monotone, and proximal ordered fuzzy ψ-contraction of type-II, \(g:A\rightarrow A \) surjective, fuzzy isometry and inverse monotone mapping. 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 \(T(A_{0}(t))\subseteq B_{0}(t) \) and \(A_{0}(t)\subseteq g(A_{0}(t))\). If there exist some elements \(x_{0}\) and \(x_{1}\) in \(A_{0}(t)\) such that
then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\). Further, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\}\in A_{0}(t)\), defined by \(M(g\overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\), converges to \(x^{\ast}\).
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
Let \(x_{0}\) and \(x_{1}\) be given points in \(A_{0}(t)\) such that
Since \(Tx_{1}\in T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\), we can choose an element \(x_{2}\in A_{0}(t)\) such that
As T is proximally monotone, we have \((gx_{1},gx_{2})\in\Delta _{(t)}\), which further implies that \((x_{1},x_{2})\in\Delta_{(t)}\). Continuing this way, we can obtain a sequence \(\{x_{n}\}\) in \(A_{0}(t)\), such that it satisfies
for each positive integer n. Having chosen \(x_{n}\), one can find a point \(x_{n+1}\) in \(A_{0}(t)\) such that
Since \(T(A_{0}(t))\subseteq B_{0}(t)\) and \(A_{0}(t)\subseteq g(A_{0}(t))\), T is proximally monotone mapping, so from (27) and (28) it follows that \((gx_{n},gx_{n+1})\in\Delta_{(t)}\) and \((x_{n},x_{n+1})\in \Delta_{(t)}\). Note that
Denote \(M(x_{n},x_{n+1},t)=\tau_{n}(t)\) for all \(t>0\), \(n\in \mathbb{N} \cup\{0\}\). The above inequality becomes
Thus \(\{\tau_{n}(t)\}\) is an increasing sequence for each \(t>0\). Consequently, \(\lim_{n\rightarrow+\infty}\tau_{n}(t)=\tau(t)\). We claim that \(\tau(t)=1\) for each \(t>0\). If not, there exist some \(t_{0}>0\) such that \(\tau(t_{0})<1\). Also, \(\tau_{n}(t_{0})\leq\tau(t_{0})\). On taking limit as \(n\rightarrow\infty\) on both sides of (30), we have \(\tau(t_{0})\leq\eta(\tau(t_{0}))<\tau(t_{0})\), a contradiction. Hence \(\tau(t)=1\) for each \(t>0\). Now we show that \(\{x_{n}\}\) is a Cauchy sequence. If not, then there exist some \(\varepsilon\in(0,1)\) and \(t_{0}>0\) such that for all \(k\in \mathbb{N} \), there are \(m_{k},n_{k}\in \mathbb{N} \), with \(m_{k}>n_{k}\geq k\) such that
If \(m_{k}\) is the least integer exceeding \(n_{k}\) and satisfying the above inequality, then
So, for all k,
On taking the limit as \(k\rightarrow\infty\) on both sides of above inequality, we obtain \(\lim_{k\rightarrow+\infty }M(x_{m_{k}},x_{n_{k}},t_{0})=1-\varepsilon\). Note that
and
imply that
From (28), we have
Thus
On taking the limit as \(k\rightarrow\infty\) in the above inequality, we get \(1-\varepsilon\leq\eta(1-\varepsilon)<1-\varepsilon\), a contradiction. Hence \(\{x_{n}\}\) is a Cauchy sequence in the closed subset \(A(t)\) of complete partially ordered fuzzy metric space \((X,M,\ast,\preceq)\). Thus there exists \(x^{\ast}\in A(t)\) such that \(\lim_{n\rightarrow\infty }M(x_{n},x^{\ast},t)=1\), for all \(t>0\). This further implies that \(M(gx^{\ast},Tx^{\ast},t)=\lim_{n\longrightarrow\infty }M(gx_{n+1},Tx_{n},t)=M(A,B,t)\) and hence \(x^{\ast}\in A_{0}(t)\) is the optimal coincidence point of a pair \(\{g,T\}\). To prove the uniqueness of \(x^{\ast}\), we show that, for any fixed element \(\overline{x}_{0}\in A_{0}(t)\), the sequence \(\{\overline{x}_{n}\}\in A_{0}(t) \) defined by \(M(g\overline{x}_{n+1},T\overline{x}_{n},t)=M(A,B,t)\) converges to \(x^{\ast }\). Suppose that there is another element \(\overline{x}_{0} \in A(t)\) such that \(0< M(x_{0},\overline{x}_{0},t)<1\) for all \(t>0\) satisfying
Suppose that \((\overline{x}_{0},x_{0})\in\Delta_{(t)}\). Then, by the given assumption, we have
a contradiction and hence the result follows. If \((\overline{x}_{0},x_{0})\notin\Delta_{(t)}\), then let \(u_{0}\) be a lower bound of \(x_{0} \) and \(\overline{x}_{0}\), and \(\overline{u}_{0}\) an upper bound of \(x_{0}\) and \(\overline{x}_{0}\). That is,
Recursively, construct sequences \(\{u_{n}\}\) and \(\{\overline{u}_{n}\}\), such that
The proximal monotonicity of the mapping T and the monotonicity of the inverse of g imply that
From \((x_{n},u_{n})\in\Delta_{(t)} \) and \((x_{n},\overline{u}_{n})\in \Delta_{(t)}\), it follows that
Hence \(\lim_{n\rightarrow\infty}\overline{x}_{n}=x^{\ast}\). □
Example 4.2
Let \(X=[-1,1]\times \mathbb{R} \) and ⪯ a usual order on \(\mathbb{R} ^{2}\). Let \(A=\{(-1,x): \mbox{for all }x\in \mathbb{R} \}\), \(B=\{(1,y): \mbox{for all }y\in \mathbb{R} \}\), and \((X,M,\ast,\preceq)\) a complete fuzzy ordered metric space as given in Example 2.17. Note that \(M(A,B,t)=\frac{t}{t+2}\), \(A_{0}(t)=A\) and \(B_{0}(t)=B\). Define \(T:A\rightarrow B\) by
Let \(g:A\rightarrow A\) be defined by \(g(-1,x)=(-1,\frac{x}{2})\). Note that g is fuzzy nonexpansive and its inverse is monotone. Obviously, \(T(A_{0}(t))\subseteq B_{0}(t)\), and \(A_{0}(t)\subseteq g(A_{0}(t))\). Note that \(u=(-1,\frac{2}{5}y_{1})\), \(v=(-1,\frac{2}{5}y_{2})\), \(x=(-1,y_{1})\), and \(y=(-1,y_{2})\in A\). Also, note that
Here \(\eta(t)=2t-t^{2}\). Thus all conditions of Theorem 4.1 are satisfied. Moreover, \((-1,0)\) is the optimal coincidence point of g and T.
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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.
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Abbas, M., Saleem, N. & De la Sen, M. Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces. Fixed Point Theory Appl 2016, 44 (2016). https://doi.org/10.1186/s13663-016-0534-3
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DOI: https://doi.org/10.1186/s13663-016-0534-3