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Optimal coincidence point results in partially ordered non-Archimedean fuzzy metric spaces

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.

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

$$ d(x,Tx)\geq d(A,B), $$

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. (1)

    ∗ is associative, commutative and continuous;

  2. (2)

    \(a\ast1=a\) for all \(a\in [0,1]\);

  3. (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:

  1. (i)

    \(M(x,y,t)>0\),

  2. (ii)

    \(x=y\) if and only if \(M(x,y,t)=1\) for all \(t>0\),

  3. (iii)

    \(M(x,y,t)=M(y,x,t)\),

  4. (iv)

    \(M(x,z,t+s)\geq M(x,y,t)\ast M(y,z,s)\) for all \(t,s>0\),

  5. (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

  1. (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

$$ B_{M}(x,\varepsilon,t)=\bigl\{ y\in X:M(x,y,t)>1-\varepsilon\bigr\} . $$

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

$$ M_{d}(x,y,t)=\frac{t}{t+d(x,y)}. $$

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:

$$\begin{aligned}& A_{0}(t) =\bigl\{ x\in A:M(x,y,t)=M(A,B,t) \mbox{ for some }y\in B \bigr\} , \\& B_{0}(t) =\bigl\{ y\in B:M(x,y,t)=M(A,B,t) \mbox{ for some }x\in A \bigr\} . \end{aligned}$$

The distance of a point \(x\in X\) from a nonempty set A for \(t>0\) is defined as

$$ M(x,A,t)=\sup_{a\in A}M(x,a,t), $$

and the distance between two nonempty sets A and B for \(t>0\) is defined as

$$ M(A,B,t)=\sup\bigl\{ M(a,b,t):a\in A,b\in B\bigr\} . $$

Definition 1.6

([4])

Let Ψ be the set of all mappings \(\psi:[0,1]\rightarrow [0,1]\) satisfying the following properties:

  1. (i)

    ψ is continuous and nondecreasing on \((0,1)\) and \(\psi (t)>t\) also \(\psi(0)=0\) and \(\psi(1)=1\).

  2. (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:

  1. (i)

    η is continuous and strictly decreasing on \((0,1)\) and \(\eta(t)< t\) for all \(t\in(0,1)\),

  2. (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

$$ f(0,x)=100(0,x). $$

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

$$ f(0,x)=\biggl(0,\frac{x}{10}\biggr). $$

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:

$$\left . \textstyle\begin{array}{r@{}} x\preceq y \\ M(u,Tx,t)=M(A,B,t) \\ M(v,Ty,t)=M(A,B,t)\end{array}\displaystyle \right \} \quad\Longrightarrow\quad M(u,v,t)\geq\psi \bigl[ M(x,y,t)\bigr], \quad \mbox{where } \psi \in\Psi. $$

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:

$$\left . \textstyle\begin{array}{r@{}} x\preceq y \\ M(u,Tx,t)=M(A,B,t) \\ M(v,Ty,t)=M(A,B,t)\end{array}\displaystyle \right \} \quad\Longrightarrow\quad M(u,v,t)\geq\psi \biggl[ M\biggl(x,y,\frac {t}{\alpha}\biggr)\biggr],\quad \mbox{where } \psi\in\Psi. $$

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:

$$\left . \textstyle\begin{array}{r@{}} x\preceq y \\ M(u,Tx,t)=M(A,B,t) \\ M(v,Ty,t)=M(A,B,t)\end{array}\displaystyle \right \} \quad\Longrightarrow \quad M(x,y,t)\leq\eta \bigl[ M(u,v,t)\bigr], \quad \mbox{where } \eta \in\Lambda. $$

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:

$$\left . \textstyle\begin{array}{r@{}} x\preceq y \\ M(u,Tx,t)=M(A,B,t) \\ M(v,Ty,t)=M(A,B,t)\end{array}\displaystyle \right \} \quad\Longrightarrow\quad u\preceq v. $$

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:

$$\left . \textstyle\begin{array}{r@{}} x\preceq y \\ M(u,Tx,t)=M(A,B,t) \\ M(v,Ty,t)=M(A,B,t)\end{array}\displaystyle \right \} \quad\Longrightarrow\quad u\succeq v. $$

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

$$ M(gx,Tx,t)=M(A,B,t) $$

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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\textit{with }(x_{0},x_{1})\in \Delta_{(t)}, $$

then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\), 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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\mbox{with }(x_{0},x_{1})\in \Delta_{(t)}. $$
(1)

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

$$ M(gx_{2},Tx_{1},t)=M(A,B,t). $$
(2)

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

$$ M(gx_{n},Tx_{n-1},t)=M(A,B,t) \quad\mbox{with }(x_{n-1},x_{n})\in \Delta_{(t)} $$
(3)

for each positive integer n. Having chosen \(x_{n}\), one can find a point \(x_{n+1}\) in \(A_{0}(t)\) such that

$$ M(gx_{n+1},Tx_{n},t)=M(A,B,t). $$
(4)

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

$$ M(x_{n},x_{n+1},t)\geq M(gx_{n},gx_{n+1},t) \geq\psi\bigl[ M(x_{n-1},x_{n},t)\bigr]. $$
(5)

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

$$ \tau_{n}(t)\geq\psi\bigl(\tau_{n-1}(t)\bigr)> \tau_{n-1}(t) $$
(6)

and

$$ \tau_{n}(t)>\tau_{n-1}(t). $$

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

$$ \tau(t_{0})\geq\psi\bigl(\tau(t_{0})\bigr)> \tau(t_{0}), $$

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

$$ M(x_{m_{k}},x_{n_{k}},t_{0})\leq1-\varepsilon. $$
(7)

Assume that \(m_{k}\) is the least integer exceeding \(n_{k}\) and satisfying the above inequality, then we have

$$ M(x_{m_{k}-1},x_{n_{k}},t_{0})>1-\varepsilon. $$
(8)

So, for all k,

$$\begin{aligned} 1-\varepsilon \geq&M(x_{m_{k}},x_{n_{k}},t_{0}) \\ \geq&M(x_{m_{k}},x_{m_{k}-1},t_{0})\ast M(x_{m_{k}-1},x_{n_{k}},t_{0}) \\ >&\tau_{m_{k}}(t_{0})\ast(1-\varepsilon). \end{aligned}$$
(9)

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

$$ M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\geq M(x_{m_{k}+1},x_{m_{k}},t_{0}) \ast M(x_{m_{k}},x_{n_{k}},t_{0})\ast M(x_{n_{k}},x_{n_{k}+1},t_{0}) $$

and

$$ M(x_{m_{k}},x_{n_{k}},t_{0})\geq M(x_{m_{k}},x_{m_{k}+1},t_{0}) \ast M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\ast M(x_{n_{k}+1},x_{n_{k}},t_{0}), $$

imply that

$$ \lim_{k\rightarrow+\infty }M(x_{m_{k}+1},x_{n_{k}+1},t_{0})=1- \varepsilon. $$

From (4), we have

$$ M(gx_{m_{k}+1},Tx_{m_{k}},t_{0})=M(A,B,t_{0}) \quad\mbox{and}\quad M(gx_{n_{k}+1},Tx_{n_{k}},t_{0})=M(A,B,t_{0}). $$

Thus

$$ M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\geq M(gx_{m_{k}+1},gx_{n_{k}+1},t_{0}) \geq \psi\bigl[ M(x_{m_{k}},x_{n_{k}},t_{0})\bigr]. $$

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

$$ M\bigl(gx^{\ast},Tx^{\ast},t\bigr)=\lim_{n\longrightarrow\infty }M(gx_{n+1},Tx_{n},t)=M(A,B,t). $$

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

$$ M(g\overline{x}_{0},T\overline{x}_{0},t)=M(A,B,t). $$
(10)

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

$$ M(\overline{x}_{0},x_{0},t)\geq M(g\overline{x}_{0},gx_{0},t) \geq\psi \bigl(M(\overline{x}_{0},x_{0},t) \bigr)>M(\overline{x}_{0},x_{0},t) $$

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,

$$ \overline{u}_{0}\succeq x_{0}\succeq u_{0} \quad\mbox{or}\quad\overline{u}_{0}\succeq \overline{x}_{0}\succeq u_{0}. $$

Recursively, construct the sequences \(\{u_{n}\}\) and \(\{\overline {u}_{n}\}\), such that

$$ M(gu_{n+1},Tu_{n},t)=M(A,B,t) \quad\mbox{and}\quad M(g \overline {u}_{n+1},T\overline{u}_{n},t)=M(A,B,t). $$

The proximal monotonicity of the mapping T and the monotonicity of the inverse of g imply that

$$ \overline{u}_{n}\succeq\overline{x}_{n}\succeq u_{n} \quad\mbox{or}\quad \overline{u}_{n}\preceq \overline{x}_{n}\preceq u_{n}. $$

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

$$ \lim_{n\rightarrow\infty}\overline{u}_{n}=\lim_{n\rightarrow\infty }u_{n}=x^{\ast}. $$

Hence

$$ \lim_{n\rightarrow\infty}\overline{x}_{n}=x^{\ast}. $$

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

$$ T(-1,x)=\biggl(1,\frac{x}{2}\biggr). $$

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

$$\begin{aligned}& M(gu,Tx,t) =M(A,B,t), \end{aligned}$$
(11)
$$\begin{aligned}& M(gv,Ty,t) =M(A,B,t). \end{aligned}$$
(12)

Also, note that

$$ M(gu,gv,t)=M\biggl(\biggl(-1,\frac{y_{1}}{2}\biggr),\biggl(-1, \frac{y_{2}}{2}\biggr),t\biggr)\geq\psi (M\bigl((-1,y_{1}),(-1,y_{2}),t \bigr)=\psi\bigl(M(x,y,t)\bigr), $$

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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\textit{with }(x_{0},x_{1})\in\Delta_{(t)}, $$

then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\). 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

$$ T(-1,x)=\biggl(1,\frac{x}{5}\biggr). $$

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

$$\begin{aligned}& M(gu,Tx,t) =M(A,B,t), \\& M(gv,Ty,t) =M(A,B,t). \end{aligned}$$

Also, note that

$$ M(gu,gv,t)=M\biggl(\biggl(-1,\frac{y_{1}}{5}\biggr),\biggl(-1, \frac{y_{2}}{5}\biggr),t\biggr)\geq\psi \bigl(M\bigl((-1,y_{1}),(-1,y_{2}),t \bigr)\bigr)=\psi\bigl(M(A,B,t)\bigr), $$

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

$$ M(x_{1},Tx_{0},t)=M(A,B,t) \quad\textit{with }(x_{0},x_{1})\in\Delta_{(t)}, $$

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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\textit{with }(x_{0},x_{1}) \in \Delta _{(t)}, $$

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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\mbox{with }(x_{0},x_{1}) \in \Delta _{(t)}. $$
(13)

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

$$ M(gx_{2},Tx_{1},t)=M(A,B,t). $$
(14)

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

$$ M(gx_{n},Tx_{n-1},t)=M(A,B,t) \quad\mbox{with }(x_{n-1},x_{n})\in\Delta_{(t)} $$
(15)

for each positive integer n. Hence after finding \(x_{n}\), we can find an element \(x_{n+1}\) in \(A_{0}(t)\) such that

$$ M(gx_{n+1},Tx_{n},t)=M(A,B,t). $$
(16)

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

$$ M(x_{n+1},x_{n},t)\geq M(gx_{n+1},gx_{n},t) \geq\psi \biggl(M\biggl(x_{n},x_{n-1},\frac{t}{\alpha}\biggr) \biggr). $$
(17)

for all \(n\geq0\). Recursively,

$$\begin{aligned} M(x_{n+1},x_{n},t) \geq&\psi\biggl(M \biggl(x_{n+1},x_{n},\frac{t}{\alpha}\biggr)\biggr)\geq \psi^{2}\biggl(M\biggl(x_{n},x_{n-1}, \frac{t}{\alpha^{2}}\biggr)\biggr)\geq \cdots \\ \geq&\psi^{n}\biggl(M\biggl(x_{1},x_{0}, \frac{t}{\alpha ^{n}}\biggr)\biggr)>M\biggl(x_{1},x_{0},\frac{t}{\alpha^{n}}\biggr), \end{aligned}$$
(18)

for all \(t>0\) and \(m,n\in \mathbb{N} \), where \(m\geq n\), so we have

$$\begin{aligned} M(x_{n},x_{m},t) \geq&M(x_{n},x_{n+1},t) \ast M(x_{n+1},x_{n+2},t)\ast M(x_{n+2},x_{n+3},t)\ast \cdots\ast M(x_{m-1},x_{m},t) \\ >&M\biggl(x_{0},x_{1},\frac{t}{\alpha^{n}}\biggr)\ast M \biggl(x_{0},x_{1},\frac {t}{\alpha ^{n+1}}\biggr)\ast\cdots\ast M \biggl(x_{0},x_{1},\frac{t}{\alpha^{m-1}}\biggr) \\ >&\prod_{i=n}^{\infty}M\biggl(x_{0},x_{1}, \frac{t}{\alpha^{i}}\biggr), \end{aligned}$$

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

$$\begin{aligned} M(gx,B,t) \geq&M(gx,Tx_{n},t) \\ \geq&M(gx,gx_{n+1},t)\ast M(gx_{n+1},Tx_{n},t) \\ =&M(gx,gx_{n+1},t)\ast M(A,B,t) \\ \geq&M(gx,gx_{n+1},t)\ast M(gx,B,t). \end{aligned}$$

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

$$ M(gx,y,t)=M(A,B,t), $$

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 \(T(A_{0}(t))\subseteq B_{0}(t)\),

$$ M(z,Tx,t)=M(A,B,t) $$
(19)

for some z in \(A(t)\). From (16) and (19) we obtain

$$ M(gx_{n+1},z,t)\geq\psi\biggl(M\biggl(x,x_{n}, \frac{t}{\alpha}\biggr)\biggr). $$
(20)

Taking the limit as \(n\rightarrow\infty\), the above inequality becomes

$$ \lim_{n\rightarrow\infty}M(gx_{n+1},z,t)\geq\lim_{n\rightarrow\infty } \psi\biggl(M\biggl(x,x_{n},\frac{t}{\alpha}\biggr)\biggr)=1, $$

which shows that \(\{gx_{n}\}\) converges to z

$$ M(gx_{n},z,t)=1. $$
(21)

Since g is continuous, the sequence \(\{gx_{n}\}\) converges to gx such that

$$ M(gx_{n},gx,t)=1. $$
(22)

Hence we have \(gx=z\),

$$ M(gx,Tx,t)=M(A,B,t)=M(z,Tx,t). $$
(23)

Suppose that there is another element \(x^{\ast}\) such that

$$ M\bigl(gx^{\ast},Tx^{\ast},t\bigr)=M(A,B,t). $$
(24)

First suppose that \((x,x^{\ast})\in\Delta_{(t)}\). From (23) and (24), it follows that

$$ M\bigl(x,x^{\ast},t\bigr)\geq M\bigl(gx,gx^{\ast},t\bigr)\geq\psi \biggl(M\biggl(x,x^{\ast},\frac {t}{\alpha}\biggr)\biggr), $$

which further implies that

$$ M\bigl(x,x^{\ast},t\bigr)>M\biggl(x,x^{\ast},\frac{t}{\alpha} \biggr), $$

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

$$ \overline{u}_{0}\succeq x_{0}\succeq u_{0} \quad\mbox{or}\quad\overline{u}_{0}\succeq \overline{x}_{0}\succeq u_{0}. $$

Recursively, we can find sequences \(\{u_{n}\}\) and \(\{\overline{u}_{n}\}\) such that

$$ M(gu_{n+1},Tu_{n},t)=M(A,B,t) \quad\mbox{and}\quad M(g \overline {u}_{n+1},T\overline{u}_{n},t)=M(A,B,t). $$

The proximal monotonicity of the mapping T and the monotonicity of the inverse of g implies that

$$ \overline{u}_{n}\succeq\overline{x}_{n}\succeq u_{n}\quad \mbox{or} \quad\overline{u}_{n}\preceq \overline{x}_{n}\preceq u_{n}. $$

Since \((x_{0},u_{0})\in\Delta_{(t)}\), also \((x_{0},\overline {u}_{0})\in \Delta_{(t)}\). It follows that

$$ \lim_{n\rightarrow\infty}\overline{u}_{n}=\lim_{n\rightarrow\infty }u_{n}=x^{\ast}. $$

Hence

$$ \lim_{n\rightarrow\infty}x_{n}=x^{\ast}. $$

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

$$ T(0,x)=\biggl(2,\frac{x}{10}\biggr). $$

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

$$\begin{aligned}[b] &M(gu,Tx,t) =M(A,B,t),\\ &M(gv,Ty,t) =M(A,B,t). \end{aligned} $$

Also, note that

$$\begin{aligned} M(gu,gv,t)&=M\biggl(\biggl(0,\frac{y_{1}}{10}\biggr),\biggl(0, \frac{y_{2}}{10}\biggr),t\biggr)\geq\psi \biggl(M\biggl((0,y_{1}),(0,y_{2}), \frac{t}{\alpha}\biggr)\biggr)\\ &=\psi\biggl(M\biggl(x,y,\frac{t}{\alpha}\biggr)\biggr), \end{aligned}$$

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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\textit{and}\quad (x_{0},x_{1})\in\Delta_{(t)}, $$

then there exists a unique element \(x^{\ast}\in A_{0}(t)\) such that \(M(gx^{\ast},Tx^{\ast},t)=M(A,B,t)\). 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

$$ M(gx_{1},Tx_{0},t)=M(A,B,t) \quad\mbox{with }(x_{0},x_{1})\in\Delta_{(t)}. $$
(25)

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

$$ M(gx_{2},Tx_{1},t)=M(A,B,t). $$
(26)

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

$$ M(gx_{n},Tx_{n-1},t)=M(A,B,t) \quad\mbox{with }(x_{n-1},x_{n})\in\Delta_{(t)}. $$
(27)

for each positive integer n. Having chosen \(x_{n}\), one can find a point \(x_{n+1}\) in \(A_{0}(t)\) such that

$$ M(gx_{n+1},Tx_{n},t)=M(A,B,t). $$
(28)

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

$$ M(x_{n},x_{n-1},t)\leq\eta\bigl[ M(gx_{n+1},gx_{n},t) \bigr]\leq\eta \bigl[ M(x_{n+1},x_{n},t)\bigr]< M(x_{n+1},x_{n},t). $$
(29)

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

$$ \tau_{n-1}(t)\leq\eta\bigl(\tau_{n}(t)\bigr)< \tau_{n}(t). $$
(30)

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

$$ M(x_{m_{k}},x_{n_{k}},t_{0})\leq1-\varepsilon. $$
(31)

If \(m_{k}\) is the least integer exceeding \(n_{k}\) and satisfying the above inequality, then

$$ M(x_{m_{k}-1},x_{n_{k}},t_{0})>1-\varepsilon. $$
(32)

So, for all k,

$$\begin{aligned} 1-\varepsilon \geq&M(x_{m_{k}},x_{n_{k}},t_{0}) \\ \geq&M(x_{m_{k}},x_{m_{k}-1},t_{0})\ast M(x_{m_{k}-1},x_{n_{k}},t_{0}) \\ >&\tau_{m_{k}}(t_{0})\ast(1-\varepsilon). \end{aligned}$$
(33)

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

$$ M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\geq M(x_{m_{k}+1},x_{m_{k}},t_{0}) \ast M(x_{m_{k}},x_{n_{k}},t_{0})\ast M(x_{n_{k}},x_{n_{k}+1},t_{0}) $$

and

$$ M(x_{m_{k}},x_{n_{k}},t_{0})\geq M(x_{m_{k}},x_{m_{k}+1},t_{0}) \ast M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\ast M(x_{n_{k}+1},x_{n_{k}},t_{0}), $$

imply that

$$ \lim_{k\rightarrow+\infty }M(x_{m_{k}+1},x_{n_{k}+1},t_{0})=1- \varepsilon. $$

From (28), we have

$$ M(gx_{m_{k}+1},Tx_{m_{k}},t_{0})=M(A,B,t_{0}) \quad\mbox{and}\quad M(gx_{n_{k}+1},Tx_{n_{k}},t_{0})=M(A,B,t_{0}). $$

Thus

$$ M(x_{m_{k}},x_{n_{k}},t_{0})\leq\eta\bigl[ M(gx_{m_{k}+1},gx_{n_{k}+1},t_{0})\bigr]\leq\eta\bigl[ M(x_{m_{k}+1},x_{n_{k}+1},t_{0})\bigr]< M(x_{m_{k}+1},x_{n_{k}+1},t_{0}). $$

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

$$ M(g\overline{x}_{0},T\overline{x}_{0},t)=M(A,B,t). $$
(34)

Suppose that \((\overline{x}_{0},x_{0})\in\Delta_{(t)}\). Then, by the given assumption, we have

$$ M(\overline{x}_{0},x_{0},t)\leq\eta\bigl(M(g \overline{x}_{0},gx_{0},t)\bigr)\leq \eta\bigl(M( \overline{x}_{0},x_{0},t)\bigr)< M(\overline{x}_{0},x_{0},t), $$

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,

$$ \overline{u}_{0}\succeq x_{0}\succeq u_{0} \quad\mbox{or}\quad\overline{u}_{0}\succeq \overline{x}_{0}\succeq u_{0}. $$

Recursively, construct sequences \(\{u_{n}\}\) and \(\{\overline{u}_{n}\}\), such that

$$ M(gu_{n+1},Tu_{n},t)=M(A,B,t) \quad\mbox{and}\quad M(g \overline {u}_{n+1},T\overline{u}_{n},t)=M(A,B,t). $$

The proximal monotonicity of the mapping T and the monotonicity of the inverse of g imply that

$$ \overline{u}_{n}\succeq\overline{x}_{n}\succeq u_{n} \quad\mbox{or}\quad \overline{u}_{n}\preceq \overline{x}_{n}\preceq u_{n}. $$

From \((x_{n},u_{n})\in\Delta_{(t)} \) and \((x_{n},\overline{u}_{n})\in \Delta_{(t)}\), it follows that

$$ \lim_{n\rightarrow\infty}\overline{u}_{n}=\lim_{n\rightarrow\infty }u_{n}=x^{\ast}. $$

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

$$ T(-1,x)=\biggl(1,\frac{x}{5}\biggr). $$

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

$$ M\bigl((-1,y_{1}),(-1,y_{2}),t\bigr)\leq\eta\biggl(M\biggl( \biggl(-1,\frac{y_{1}}{5}\biggr),\biggl(-1,\frac {y_{2}}{5}\biggr),t\biggr) \biggr). $$

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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