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) ∗ 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

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

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:

• (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

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

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.