# 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 [27] 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 $$\{(x,y)\in A_{0}(t)\times A_{0}(t): \mbox{either } x\preceq y\mbox{ or }{y\preceq x} \}$$.

### 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.

## References

1. Fan, K: Extensions of two fixed point theorems of F. E. Browder. Math. Z. 112, 234-240 (1969)

2. Sankar Raj, V, Veeramani, P: Best proximity pair theorems for relatively nonexpansive mappings. Appl. Gen. Topol. 10, 21-28 (2009)

3. Sadiq Basha, S, Shahzad, N, Jeyaraj, R: Optimal approximate solutions of fixed point equations. Abstr. Appl. Anal. 2011, 174560 (2011)

4. Saleem, N, Ali, B, Abbas, M, Raza, Z: Fixed points of Suzuki type generalized multivalued mappings in fuzzy metric spaces with applications. Fixed Point Theory Appl. 2015, 36 (2015)

5. Shahzad, N, Sadiq Basha, S, Jeyaraj, R: Common best proximity points: global optimal solutions. J. Optim. Theory Appl. 148, 69-78 (2011)

6. Suzuki, T, Kikkawa, M, Vetro, C: The existence of best proximity points in metric spaces with the property UC. Nonlinear Anal. 71, 2918-2926 (2009)

7. Vetro, C: Best proximity points: convergence and existence theorems for p-cyclic mappings. Nonlinear Anal. 73, 2283-2291 (2010)

8. Zadeh, LA: Fuzzy sets. Inf. Control 8, 103-112 (1965)

9. Kramosil, I, Michalek, J: Fuzzy metric and statistical metric spaces. Kybernetika 11, 326-334 (1975)

10. George, A, Veeramani, P: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64, 395-399 (1994)

11. George, A, Veeramani, P: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90, 365-368 (1997)

12. Vetro, C, Salimi, P: Best proximity point results in non-Archimedean fuzzy metric spaces. Fuzzy Inf. Eng. 4, 417-429 (2013)

13. Sadiq Basha, S: Best proximity point theorems on partially ordered sets. Optim. Lett. 7, 1035-1043 (2013)

14. Schweizer, B, Sklar, A: Statistical metric spaces. Pac. J. Math. 10, 314-334 (1960)

15. Grabiec, M: Fixed points in fuzzy metric spaces. Fuzzy Sets Syst. 27, 385-389 (1983)

16. Gutierrez Garcia, J, Romaguera, S: Examples of non-strong fuzzy metrics. Fuzzy Sets Syst. 162, 91-93 (2011)

17. Gregori, V, Sapena, A: On fixed-point theorems in fuzzy metric spaces. Fuzzy Sets Syst. 125, 245-252 (2002)

18. Hosseini, BH, Saadati, R, Amini, M: Alexandroff theorem in fuzzy metric spaces. Math. Sci. Res. J. 8(12), 357-361 (2004)

19. Altun, I, Mihet, D: Ordered non-Archimedean fuzzy metric spaces and some fixed point results. Fixed Point Theory Appl. 2010, 782680 (2010). doi:10.1155/2010/782680

## 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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Correspondence to Manuel De la Sen.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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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

• 47H10
• 47H04
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### Keywords

• fuzzy metric space
• monotone fuzzy proximal contraction
• fuzzy ordered proximal contractions
• fuzzy expansive
• fuzzy nonexpansive
• s-increasing sequence
• t-norm