### Definition 2.1

Let *A* be a nonempty subset of a non-Archimedean fuzzy metric space \((X,M,\ast)\). A self mapping *f* on *A* is said to be (a) fuzzy isometry if \(M(fx,fy,t)=M(x,y,t)\) for all \(x,y\in A \) and \(t>0\) (b) fuzzy expansive if, for any \(x,y\in A \) and \(t>0\), we have \(M(fx,fy,t)\leq M(x,y,t)\), (c) fuzzy nonexpansive if, for any \(x,y\in A \) and \(t>0\), we have \(M(fx,fy,t)\geq M(x,y,t)\).

### Example 2.2

Let \(X=[0,1]\times \mathbb{R} \) and \(d:X\times X\rightarrow \mathbb{R} \) be a usual metric on *X*. Let \(A=\{(0,x):x\in \mathbb{R} \}\). Note that \((X,M_{d},\cdot)\) is non-Archimedean fuzzy metric space, where \(M_{d}\) is standard fuzzy metric induced by *d*. Define the mapping \(f:A\rightarrow A\) by \(f(0,x)=(0,-x)\). Note that \(M_{d}(w,u,t)=\frac{t}{ t+\vert x-y\vert }=M(fw,fu,t)\), where \(w=(0,x)\), \(u=(0,y)\in A\).

Note that every fuzzy isometry is fuzzy expansive but the converse does not hold in general.

### Example 2.3

Let \(X=[0,4]\times \mathbb{R} \) and \(d:X\times X\rightarrow \mathbb{R} \) be a usual metric on *X*. Let \(A=\{(0,x):x\in \mathbb{R} \}\). Note that \((X,M_{d},\cdot)\) is a non-Archimedean fuzzy metric space, where \(M_{d}\) is the standard fuzzy metric induced by *d*. Define the mapping \(f:A\rightarrow A\) by

If \(x=(0,0)\) and \(y=(0,4) \) then \(M(x,y,t)=\frac{t}{t+4} \) and \(M(fx,fy,t)=\frac{t}{t+400}\). This shows that *f* is fuzzy expansive but not a fuzzy isometry.

### Example 2.4

Let \(X=[0,1]\times \mathbb{R} \), \(d:X\times X\rightarrow \mathbb{R} \) a usual metric on *X* and \(A=\{(0,x):x\in \mathbb{R} \}\). Define a mapping \(f:A\rightarrow A\) by

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

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*). □