# Common Fixed Point Theorems on Complete and Weak $G$-Complete Fuzzy Metric
Spaces

Sugata Adhya and A. Deb Ray  Department of Mathematics, The Bhawanipur
Education Society College. 5, Lala Lajpat Rai Sarani, Kolkata 700020, West
Bengal, India<EMAIL_ADDRESS>Department of Pure Mathematics,
University of Calcutta. 35, Ballygunge Circular Road, Kolkata 700019, West
Bengal, India<EMAIL_ADDRESS>
###### Abstract.

Motivated by Gopal and Vetro [Iranian Journal of Fuzzy Systems, 11(3),
95-107], we introduce a symmetric pair of $\beta$-admissible mappings and
obtain common fixed point theorems for such a pair in complete and weak
$G$-complete fuzzy metric spaces. In particular, we rectified, generalize and
improve the common fixed point theorem obtained by Turkoglu and Sangurlu
[Journal of Intelligent & Fuzzy Systems, 26(1), 137-142] for two fuzzy
$\psi$-contractive mappings. We include non-trivial examples to exhibit the
generality and demonstrate our results.

AMS Subject Classification: 47H10, 54A40, 54H25.
Keywords: Common fixed point, Fuzzy metric space, Weak $G$-complete.

##  1\. Introduction

Finding an appropriate analogue for metric spaces in fuzzy setting was a long
standing problem. In 1975, motivated by the idea of Menger spaces [11],
Kramosil and Michalek [10] gave a solution to this problem and introduced
fuzzy metric spaces. Grabiec [5], in 1988, defined Cauchy sequences in such
spaces. However, while modifying the definition of fuzzy metric, George and
Veeramani [3] strengthened Grabiec’s definition of Cauchy sequences which is
now widely accepted as standard for fuzzy metric spaces.

In fact, Grabiec’s original definition of Cauchy sequence (now known as
$G$-Cauchy sequence [7]) was so weak that even a compact fuzzy metric space
fails to be complete (now known as $G$-complete [7]) in the Grabiec’s sense.
Due to this drawback, in [7], Gregori et. al. introduced a new form of
completeness. It is called weak $G$-completeness for fuzzy metric spaces. Weak
$G$-completeness has been further studied in [1] and [2]. In this paper we
establish certain fixed point theorems in weak $G$-complete fuzzy metric
spaces.

It is known that, alike metric spaces, fixed point theory is a rich subfield
of fuzzy metric spaces where contractive and contractive-type mappings play
important roles for obtaining fixed points theorems. The first fuzzy version
of Banach Contraction Principle was established in 1988 by Grabiec for
$G$-complete fuzzy metric spaces [5]. In 2002, Gregori and Sapena introduced
fuzzy contractive mappings and obtained several fixed point theorems for
complete fuzzy metric spacesi [8]. Mihet enlarged this class of contractive
mappings and introduced the notion of fuzzy $\psi$-contractive mappings. This
new class of mappings was utilized to establish a new version of fuzzy Banach
contraction theorem for complete non-Archimedean fuzzy metric spaces [12]
which was further generalized for weak $G$-complete fuzzy metric spaces [7].

The above class of contractive mappings, introduced by Mihet, has been
extensively used to obtain fixed point theorems in fuzzy metric spaces. In
2014, Turkoglu and Sangurlu obtained a common fixed point theorem for a pair
of fuzzy $\psi$-contractive mappings in $G$-complete fuzzy metric spaces [15].

In this paper, motivated by the work of Gopal and Vetro [4], we introduce a
symmetric pair of $\beta$-admissible mappings and a pair of
$\beta$-$\psi$-fuzzy contractive mappings. These new families are utilized
here to establish common fixed point theorems in complete and weak
$G$-complete fuzzy metric spaces, both in the sense of [3] and [10]. In
particular, we rectified the fixed point theorem obtained by Turkoglu et. al.
[15] and substantially generalize and improve it. Our theory is supported and
illustrated by appropriate examples.

##  2\. Preliminaries

In this section, we recall some basic definitions and facts which are referred
subsequently.

######  Definition 1.

[14] A mapping $*:[0,1]\times[0,1]\to[0,1]$ is called a continuous $t$-norm if
(i) $*$ is associative and commutative, (ii) $*$ is continuous, (iii)
$a*1=a,~{}\forall~{}a\in[0,1],$ and (iv) for $a,b,c,d\in[0,1],$ $a\leq c,b\leq
d\implies a*b\leq c*d.$

It is easy to note that, the followings are examples of continuous $t$-norms:

(i) $a*b=\min(a,b),$ and

(ii) $a*b=ab$

for any $a,b\in[0,1],$

######  Definition 2.

(Kramosil and Michalek [10]) Given a nonempty set $X,$ a continuous $t$-norm
$*$ and a mapping $M:X\times X\times[0,\infty)\to[0,1],$ the ordered triple
$(X,M,*)$ is called a KM fuzzy metric space if, for all $x,y,z\in X$ and
$s,t>0,$ the following conditions hold:

a) $M(x,y,0)=0,$

b) $M(x,y,t)=1,~{}\forall~{}t>0\iff x=y,$

c) $M(x,y,t)=M(y,x,t),$

d) $M(x,y,t)*M(y,z,s)\leq M(x,z,t+s),$

e) $M(x,y,.):[0,\infty)\to[0,1]$ is left continuous.

######  Definition 3.

(George and Veeramani [3]) Given a nonempty set $X,$ a continuous $t$-norm $*$
and a mapping $M:X\times X\times(0,\infty)\to[0,1],$ the ordered triple
$(X,M,*)$ is called a GV fuzzy metric space if, for all $x,y,z\in X$ and
$s,t>0,$ the following conditions hold:

a) $M(x,y,t)>0,$

b) $M(x,y,t)=1\iff x=y,$

c) $M(x,y,t)=M(y,x,t),$

d) $M(x,y,t)*M(y,z,s)\leq M(x,z,t+s),$

e) $M(x,y,.):(0,\infty)\to[0,1]$ is continuous.

Unless otherwise specified, by a fuzzy metric space we refer to the GV fuzzy
metric space.

######  Definition 4.

[9] A (KM) fuzzy metric space $(X,M,*)$ is said to be non-Archimedean if
$M(x,y,t)*M(y,z,s)\leq M(x,z,\max\\{t,s\\}),$ for all $x,y,z\in X$ and
$s,t>0.$

######  Lemma 1.

([3], [5]) Given a (KM) fuzzy metric space $(X,M,*),$ $M(x,y,\cdot)$ defines a
nondecreasing map, $\forall~{}x,y\in X.$

Let $(X,M,*)$ be a (KM) fuzzy metric space. It is well-known ([3], [7]) that,
$\\{B(x,r,t):x\in X,r\in(0,1),t>0\\}$ forms a base for some topology
$\tau_{M}$ on $X,$ where $B(x,r,t)=\\{y\in X:M(x,y,t)>1-r\\},~{}\forall~{}x\in
X,r\in(0,1),t>0$. The topological behaviour of $(X,M,*)$ are defined with
respect to the topology $\tau_{M}.$ In particular, given two (KM) fuzzy metric
spaces $(X,M,*)$ and $(Y,N,\star),$ a mapping $f:X\to Y$ is called continuous
if $f$ is continuous as a mapping from $(X,\tau_{M})$ to $(Y,\tau_{N}).$

Similarly, sequential convergence are defined as follows: A sequence $(x_{n})$
in a (KM) fuzzy metric space $(X,M,*)$ is said to be convergent to some $x\in
X$ (resp. clusters), if it does so in $(X,\tau_{M}).$

It is easy to note that, if $f$ is a continuous mapping from a (KM) fuzzy
metric space $(X,M,*)$ to a (KM) fuzzy metric space $(Y,N,\star),$ and
$(x_{n})$ is a sequence in $X$ converging to $x\in X,$ then $f(x_{n})\to f(x)$
as $n\to\infty.$

The following is an easy consequence that has been shown in [3] for GV fuzzy
metric spaces. The case for KM fuzzy metric spaces is similar as stated next.

######  Theorem 2.1.

A sequence $(x_{n})$ in a (KM) fuzzy metric space $(X,M,*)$ converges to $x\in
X$ if and only if $\lim\limits_{n\to\infty}M(x_{n},x,t)=1,~{}\forall~{}t>0.$

######  Definition 5.

([3], [13]) Let $(X,M,*)$ be a (KM) fuzzy metric space. A sequence $(x_{n})$
in $X$ is called Cauchy if for $\epsilon\in(0,1),t>0,$ there exists
$k\in\mathbb{N}$ such that $M(x_{m},x_{n},t)>1-\epsilon,~{}\forall~{}m,n\geq
k.$ Clearly every convergent sequence in $(X,M,*)$ is Cauchy. $(X,M,*)$ is
complete if every Cauchy sequence in it converges.

######  Definition 6.

([5], [7]) Let $(X,M,*)$ be a (KM) fuzzy metric space. A sequence $(x_{n})$ in
$X$ is called $G$-Cauchy if $\lim\limits_{n\to\infty}M(x_{n},$
$x_{n+1},t)=1,~{}\forall~{}t>0.$ If, in $(X,M,*),$ every $G$-Cauchy sequence
converges, then $(X,M,*)$ is said to be $G$-complete.

Though $G$-completeness necessarily imply completeness in (KM) fuzzy metric
spaces, a compact (KM) fuzzy metric space may not be $G$-complete. To overcome
this drawback, the following weaker version of completeness has been
introduced in [7] .

A (KM) fuzzy metric space in which every $G$-Cauchy sequence clusters is
called a weak $G$-complete (KM) fuzzy metric space.

We finish this section by recalling the definitions of $\beta$-$\psi$-fuzzy
contractive mapping and $\beta$-admissible mapping introduced by Gopal and
Vetro in [4].

Following [12], we denote by $\Psi$ the family of all mappings
$\psi:[0,1]\to[0,1]$ such that,

(i) $\psi$ is non-decreasing and continuous,

(ii) $\psi(t)>t,~{}\forall~{}t\in(0,1).$

It is easy to check that $\psi(1)=1$ and
$\lim\limits_{n\to\infty}\psi^{n}(r)=1,~{}\forall~{}\psi\in\Psi,r\in(0,1)$
(e.g. consult [16]).

######  Definition 7.

Let $(X,M,*)$ be a (KM) fuzzy metric space and $f$ a self-mapping on $X$. For
some $\psi\in\Psi,$ and a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$
$f$ is called a $\beta$-$\psi$-fuzzy contractive mappings if $\forall~{}x,y\in
X$ with $x\neq y$ and $t>0,$

$M(x,y,t)>0\implies\beta(x,y,t)M(fx,fy,t)\geq\psi(M(x,y,t)).$

######  Definition 8.

Let $(X,M,*)$ be a (KM) fuzzy metric space and $f$ a self-mapping on $X$. For
a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $f$ is called
$\beta$-admissible if $\forall~{}x,y\in X,t>0,$

$\beta(x,y,t)\leq 1\implies\beta(fx,fy,t)\leq 1.$

##  3\. Main Results

We begin this section by introducing a pair of $\beta$-$\psi$-fuzzy
contractive mappings and a symmetric pair of $\beta$-admissible mappings that
extend respectively the class of $\beta$-$\psi$-fuzzy contractive mappings and
$\beta$-admissible mappings in a (KM) fuzzy metric space.

######  Definition 9.

([12], [16]) Let $(X,M,*)$ be a (KM) fuzzy metric space and $\psi\in\Psi.$

a) A mapping $f:X\to X$ is called fuzzy $\psi$-contractive if
$\forall~{}x,y,\in X,t>0,$

$M(x,y,t)>0\implies M(fx,fy,t)\geq\psi(M(x,y,t)).$

b) Given a pair of mappings $f,g:X\to X,$ $(f,g)$ is called a pair of fuzzy
$\psi$-contractive mappings if $\forall~{}x,y,\in X,t>0,$

$M(x,y,t)>0\implies
M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}).$

######  Definition 10.

Let $(X,M,*)$ be a (KM) fuzzy metric space and $f,g$ be self-mappings on $X$.
For a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$ the pair $(f,g)$ is
called a symmetric pair of $\beta$-admissible mappings if $\forall~{}x,y\in
X,t>0,$

$\beta(x,y,t)\leq
1\implies\max\\{\beta(fx,gy,t),\beta(gy,fx,t),\beta(gx,fy,t),\beta(fy,gx,t)\\}\leq
1.$

We note that if $\beta(x,y,t)\equiv 1,$ then a pair of $\beta$-$\psi$-fuzzy
contractive mappings is a pair of $\psi$-contractive mappings.

######  Theorem 3.1.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
continuous self-mappings on $X$ satisfying $M(x,fx,t),M(x,gx,t)>0,$
$\forall~{}x\in X,t>0.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,g)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(x_{0},fx_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Define a sequence $(x_{n})$ as follows:
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$.

We have $\beta(x_{0},x_{1},t)=\beta(x_{0},fx_{0},t)\leq 1,$ $\forall~{}t>0.$
Since $\beta(x_{n-1},x_{n},t)\leq
1\implies\beta(x_{n},x_{n+1},t)\leq\max\\{\beta(fx_{n-1},gx_{n},t),\beta(gx_{n-1},fx_{n},t)\\}\leq
1,$ $\forall~{}n\in\mathbb{N},t>0,$ so by applying induction, we obtain
$\beta(x_{n},x_{n+1},t)\leq 1,~{}\forall~{}n=0,1,2,\cdots$ and $t>0.$

Consequently, $\beta(x_{n+1},x_{n},t)\leq 1,~{}\forall~{}n\in\mathbb{N},t>0.$

Let $n$ be a positive integer.

If $n$ is odd, then for chosen $t>0,$
$M(x_{n-1},x_{n},t)>0\implies M(x_{n},x_{n+1},t)=M(fx_{n-1},gx_{n},t)\\\
\geq\beta(x_{n-1},x_{n},t)M(fx_{n-1},gx_{n},t)\\\
\geq\psi(\min\\{M(x_{n-1},x_{n},t),M(x_{n-1},fx_{n-1},t),M(x_{n},gx_{n},t)\\})\\\
=\psi(M(x_{n-1},x_{n},t))>0.$

Again if $n$ is $\text{even},$ then for chosen $t>0,$
$M(x_{n-1},x_{n},t)>0\implies M(x_{n},x_{n+1},t)=M(gx_{n-1},fx_{n},t)\\\
=M(fx_{n},gx_{n-1},t)\\\ \geq\beta(x_{n},x_{n-1},t)M(fx_{n},gx_{n-1},t)\\\
\geq\psi(\min\\{M(x_{n},x_{n-1},t),M(x_{n},fx_{n},t),M(x_{n-1},gx_{n-1},t)\\})\\\
=\psi(M(x_{n-1},x_{n},t))>0.$

Thus, by using induction, we have
$M(x_{n},x_{n+1},t)\geq\psi(M(x_{n-1},x_{n},t)),$
$\forall~{}n\in\mathbb{N},t>0$ and hence,
$M(x_{n},x_{n+1},t)\geq\psi^{n}(M(x_{0},x_{1},t)),$
$\forall~{}n\in\mathbb{N},t>0.$

Thus $\lim\limits_{n\to\infty}M(x_{n},x_{n+1},t)=1,~{}\forall~{}t>0.$ Hence
$(x_{n})$ is a $G$-Cauchy sequence in $X$.

Since $(X,M,*)$ is weak $G$-complete, $(x_{n})$ has a cluster point $x$ in
$X.$

So there exists a subsequence $(x_{r_{n}})$ of $(x_{n})$ such that
$(x_{r_{n}})$ converges to $x$ where $r_{n}$’s are either all even or all odd.

Without loss of generality, suppose all the $r_{n}$’s are even. Then
$\forall~{}n\in\mathbb{N},t>0,$

$M(x,fx,t)\geq
M\left(x,x_{r_{n}},\frac{t}{3}\right)*M\left(x_{r_{n}},x_{r_{n}+1},\frac{t}{3}\right)*M\left(x_{r_{n}+1},fx,\frac{t}{3}\right)$

$=M\left(x,x_{r_{n}},\frac{t}{3}\right)*M\left(x_{r_{n}},x_{r_{n}+1},\frac{t}{3}\right)*M\left(fx_{r_{n}},fx,\frac{t}{3}\right)$

Taking limit as $n\to\infty,$ we have $M(x,fx,t)=1,~{}\forall~{}t>0,$ since
$f$ is continuous and $(x_{n})$ is $G$-Cauchy. Hence $fx=x.$

Again, since $\forall~{}t>0,$ $M(x_{r_{n}-1},x,t)\geq
M\left(x_{r_{n}-1},x_{r_{n}},\frac{t}{2}\right)*M\left(x_{r_{n}},x,\frac{t}{2}\right)\to
1*1=1$ as $n\to\infty,$ so $\forall~{}t>0,$ $M(x_{r_{n}-1},x,t)\to 1$ as
$n\to\infty,$ and consequently, $\forall~{}t>0,$ $M(x,gx,t)\geq
M\left(x,x_{r_{n}},\frac{t}{2}\right)*M\left(x_{r_{n}},gx,\frac{t}{2}\right)=M\left(x,x_{r_{n}},\frac{t}{2}\right)*M\left(gx_{r_{n}-1},gx,\frac{t}{2}\right)\to
1*1=1$ as $n\to\infty,$ since $g$ is continuous.

Thus $M(x,gx,t)=1,~{}\forall~{}t>0,$ and hence $gx=x.$

Again, if all the $r_{n}$’s are odd, it can be similarly shown that $fx=gx=x.$

Thus, in general, $x$ is a common fixed point of both $f$ and $g.$

Let us now assume, in addition to condition (i) and (ii), that
$\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for $x,y~{}(x\neq y)\in X,$
$M(x,y,t)>0,$ $\forall~{}t>0.$

If possible, let $x,y$ be two common fixed points of $f$ and $g$ such that
$x\neq y.$ Then there exists $t_{0}>0$ such that $0<M(x,y,t_{0})<1.$

Now $M(x,y,t_{0})\\\ \geq\beta(x,y,t_{0})M(fx,gy,t_{0})\\\
\geq\psi(\min\\{M(x,y,t_{0}),M(x,fx,t_{0}),M(y,gy,t_{0})\\})\\\
=\psi(M(x,y,t_{0}))\\\ >M(x,y,t_{0}),$ a contradiction.

Hence the common fixed point is unique. ∎

By putting $\beta\equiv 1$ in Theorem 3.1, we have the following:

######  Corollary 3.1.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
continuous self-mappings on $X$ satisfying $M(x,fx,t),M(x,gx,t)>0,$
$\forall~{}x\in X,t>0.$ Suppose for some $\psi\in\Psi$ and $x_{0}\in X,$
$M(x_{0},fx_{0},t)>0,$ $\forall~{}t>0$ and
$M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$
$\forall~{}x,y\in X,t>0.$ Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if for $x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then
the fixed point is unique.

The last corollary shows that in the following common fixed point result
obtained by Turkoglu and Sangurlu, we do not require to specify the continuous
$t$-norm $*$ or assume $f,g$ to be fuzzy $\psi$-contractive mappings.

######  Theorem 3.2.

([15], Common fixed point theorem by Turkoglu and Sangurlu) Let $(X,M,*)$ be a
$G$-complete KM fuzzy metric space such that $a*b=\min\\{a,b\\}.$ Let there is
a $x_{0}\in X$ such that $M(x_{0},fx_{0},t)>0,~{}\forall~{}t>0.$ If $f$ and
$g$ are continuous self-mappings on $X$ satisfying $M(x,fx,t),M(x,gx,t)>0,$
$\forall~{}x\in X,t>0$ such that for some $\psi\in\Psi,$

(i) $f,g$ are fuzzy $\psi$-contractive mappings,

(ii)
$M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),~{}\forall~{}x,y\in
X,t>0,$

Then $f$ and $g$ have a common fixed point in $X.$

######  Note 1.

It should be noted, in [15], Turkoglu et. al. included the following
additional condition in the hypothesis:

$(x_{n})$ is a sequence in $X$ such that
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=f(x_{2n}),x_{2n+2}=g(x_{2n+1}),\cdots.$

However, noting that such sequence always exists, we remove it from the
statement of Theorem 3.2.

Moreover, we have included the additional condition:

$M(x,fx,t),M(x,gx,t)>0,$ $\forall~{}x\in X,t>0$

in the hypothesis of Theorem 3.2 noting that it is essential even for the
proof provided by Turkoglu et. al. [15].

######  Note 2.

In [15], Turkoglu et. al. concluded that under the hypothesis of Theorem 3.2,
$f$ and $g$ have a unique common fixed point. However this is not true as is
exhibited next.

Consider the KM fuzzy metric space $(X,M,*),$ where $X=[0,1],$
$a*b=\min\\{a,b\\},~{}\forall~{}a,b\in[0,1],$ and for $x,y\in X,$

$M(x,y,t)=\begin{cases}0&\text{if }x\neq y,t\geq 0\\\ \vskip 9.0pt\cr
0&\text{if }x=y,t=0\\\ \vskip 9.0pt\cr 1&\text{if }x=y,t>0\end{cases}$

Then $X$ is a $G$-complete KM fuzzy metric space.

Then by setting $\psi(t)=\sqrt{t},~{}\forall~{}t\in[0,1],$ for $f,g:X\to X$
given by $f(x)=g(x)=x,~{}\forall~{}x\in X$ the hypothesis of Theorem 3.2 get
satisfied. However every point of $X$ is a common fixed point of $f$ and $g$
and hence the common fixed point is not unique.

In what follows, we exhibit the applicability of our theorem, namely, Theorem
3.1 over Theorem 3.2 (due to [15]) in the following aspects:

(a) Theorem 3.1 is applicable even for a weak $G$-complete (KM) fuzzy metric
space $(X,M,*)$ which is not $G$-complete;

(b) Theorem 3.1 is applicable also for $\beta\not\equiv 1;$

(c) Theorem 3.1 is applicable even when $a*b$ is not defined as
$\min\\{a,b\\}.$

######  Example 3.1.

Consider the fuzzy metric space $(X,M,\cdot),$ where
$X=\left\\{\frac{1}{2^{n}}:n\geq 2\right\\}$
$\bigcup\left[\frac{1}{2},1\right],$
$M(x,y,t)=\frac{\min\\{x,y\\}}{\max\\{x,y\\}},~{}\forall~{}x,y\in X,t>0$ and
$\cdot$ is the usual product of reals. It is known that $(X,M,\cdot)$ is a
weak $G$-complete fuzzy metric space which is not $G$-complete [7]. Further,
$\tau_{M}$ defines the usual topology of $\mathbb{R}$ restricted to the set
$X$ [6].

Let $f,g$ be two self-mappings on $X$ such that $fx=x,$
$gx=1,~{}\forall~{}x\in X.$ Then $f,g$ are continuous on $X.$

Now by setting $\beta:X^{2}\times(0,\infty)\to(0,\infty)$ as

$\beta(x,y,t)=\begin{cases}\frac{1}{x}&\text{if }x\leq y\\\ \vskip
9.0pt\cr\frac{1+x}{2x}&\text{if }y<x<1\\\ \vskip 9.0pt\cr 2&\text{if
}y<x=1\end{cases}$

for all $x,y\in X,t>0,$ we see that $(f,g)$ defines a symmetric pair of
$\beta$-admissible mappings and $\forall~{}t>0,$ $\beta(x_{0},fx_{0},t)\leq
1,M(x_{0},fx_{0},t)>0$ hold for $x_{0}=1.$

Let $\psi:[0,1]\to[0,1]$ be defined by
$\psi(x)=\frac{1+x}{2},~{}\forall~{}x\in[0,1].$ Clearly, $\psi\in\Psi.$

We now show that, $(f,g)$ is a pair of $\beta$-$\psi$-fuzzy contractive
mappings.

Choose $x,y\in X$ and $t>0.$

Case I: $x\leq y.$ Then $\beta(x,y,t)M(fx,gy,t)=\frac{1}{x}\times
M(x,1,t)=1\\\ \geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}).$

Case II: $y<x<1.$ Then $\beta(x,y,t)M(fx,gy,t)=\frac{1+x}{2}$ and

$\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\})=\psi(\min\\{\frac{y}{x},1,y\\})=\psi(y)=\frac{1+y}{2}.$

Since $x>y,$ we obtain

$\beta(x,y,t)M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}).$

Case III: $y<x=1.$ Then $\beta(x,y,t)M(fx,gy,t)=2\times
M(x,1,t)\geq\psi(\min\\{M(x,y,t),\\\ M(x,fx,t),M(y,gy,t)\\}).$

Thus, in view of Theorem 3.1, $f,g$ have a common fixed point.

In what follows we show that, in Theorem 3.1, the last condition is essential
to ensure the uniqueness of the common fixed point.

######  Example 3.2.

Consider the fuzzy metric space $(X,M,\cdot)$ of Example 3.1. Let $f,g$ be
self-mappings on $X$ such that

$fx=gx=\begin{cases}\frac{1}{4}&\text{if }x=\frac{1}{4}\\\ \vskip 9.0pt\cr
1&\text{otherwise}\end{cases}.$

Clearly, $f,g$ are continuous on $X.$

If $\beta:X^{2}\times(0,\infty)\to(0,\infty)$ is defined by

$\beta(x,y,t)=\begin{cases}1&\text{if }x=y=1\\\ \vskip 9.0pt\cr
4&\text{otherwise}\end{cases}$

then $(f,g)$ is a pair of $\beta$-$\psi$-fuzzy contractive mappings,
$\forall~{}\psi\in\Psi.$

It is easy to see that conditions (i) and (ii) of Theorem 3.1 hold
immediately.

So, in view of Theorem 3.1, $f,g$ have a common fixed point.

However the fixed point is not unique. Indeed $x=1$ and $x=\frac{1}{4}$ are
two such points.

######  Corollary 3.2.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f$ be a
continuous self-mapping on $X$ satisfying $M(x,fx,t)>0,$ $\forall~{}x\in
X,t>0.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,f)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,f)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(x_{0},fx_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

######  Theorem 3.3.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
continuous self-mappings on $X$ satisfying $M(x,fx,t),M(x,gx,t)>0,$
$\forall~{}x\in X,t>0.$ Suppose for some $\psi_{1},\psi_{2},\psi_{3}\in\Psi,$
and a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(fx,gy,t)\geq\psi_{1}(M(x,y,t))+\psi_{2}(M(x,fx,t))+\psi_{3}(M(y,gy,t)),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(x_{0},fx_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Set $\psi=\min\\{\psi_{1},\psi_{2},\psi_{3}\\}.$ Then it is easy to see that
$\psi\in\Psi.$

Further $\forall~{}x,y\in X,t>0,$

$M(x,y,t)>0\implies\beta(x,y,t)M(fx,gy,t)\\\
\geq\psi_{1}(M(x,y,t))+\psi_{2}(M(x,fx,t))+\psi_{3}(M(y,gy,t))\\\
\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}).$

Thus the conclusion follows from Theorem 3.1. ∎

######  Corollary 3.3.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
continuous self-mappings on $X$ satisfying $M(x,fx,t),M(x,gx,t)>0,$
$\forall~{}x\in X,t>0.$ Suppose for some $a,b,c>1,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(fx,gy,t)\geq aM(x,y,t)+bM(x,fx,t)+cM(y,gy,t),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(x_{0},fx_{0},t),$ $\forall~{}t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Immediate from Theorem 3.3 by setting

$\psi_{1}(x)=\min\\{ax,\sqrt{x}\\},$

$\psi_{2}(x)=\min\\{bx,\sqrt{x}\\},$

$\psi_{3}(x)=\min\\{cx,\sqrt{x}\\},$

$\forall~{}x\in[0,1].$ ∎

In the next theorem, we omit the continuity hypothesis of $f,g$ in Theorem 3.1
for fuzzy metric spaces defined by George and Veeramani [3].

######  Theorem 3.4.

Let $(X,M,*)$ be a non-Archimedean weak $G$-complete fuzzy metric space and
$f,g$ be self-mappings on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,g)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and a subsequence $(x_{r_{n}})$ of
$(x_{n})$ with $\lim\limits_{n\to\infty}x_{r_{n}}=x,$ we have
$\beta(x_{r_{n}},x,t)\leq 1,~{}\forall~{}n\in\mathbb{N},t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0,$ then the fixed
point is unique.

###### Proof.

Proceeding as in Theorem 3.1, we obtain the $G$-Cauchy sequence $(x_{n})$ in
$X$ given by
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$
such that $\beta(x_{n},x_{n+1},t)\leq 1,~{}\forall~{}n=0,1,2,\cdots$ and
$t>0.$

Since $(X,M,*)$ is weak $G$-complete, there exists a subsequence $(x_{r_{n}})$
of $(x_{n})$ such that $(x_{r_{n}})$ converges to some $x\in X$ where
$r_{n}$’s are either all even or all odd.

Without loss of generality, suppose all the $r_{n}$’s are even.

If possible, let $gx\neq x.$ Then $0<M(x,gx,t_{0})<1,$ for some $t_{0}>0.$

Since $X$ is non-Archimedean, $\forall~{}n\in\mathbb{N}$ we have

$M(x,gx,t_{0})\geq M(x,x_{r_{n}+1},t_{0})*M(x_{r_{n}+1},gx,t_{0})\\\
=M(x,x_{r_{n}+1},t_{0})*M(fx_{r_{n}},gx,t_{0})\\\ \geq
M(x,x_{r_{n}+1},t_{0})*(\beta(x_{r_{n}},x,t_{0})M(fx_{r_{n}},gx,t_{0}))\text{(by
(iii))}\\\ \geq
M(x,x_{r_{n}+1},t_{0})*\psi(\min\\{M(x_{r_{n}},x,t_{0}),M(x_{r_{n}},fx_{r_{n}},t_{0}),M(x,gx,t_{0})\\})\\\
\geq
M(x,x_{r_{n}},t_{0})*M(x_{r_{n}},x_{r_{n}+1},t_{0})*\psi(\min\\{M(x_{r_{n}},x,t_{0}),M(x_{r_{n}},x_{r_{n}+1},t_{0}),M(x,gx,t_{0})\\})\\\
\to 1*1*\psi(M(x,gx,t_{0}))=\psi(M(x,gx,t_{0}))$ as $n\to\infty,$ since $*$ is
continuous and $(x_{n})$ is $G$-Cauchy.

Thus $M(x,gx,t_{0})\geq\psi(M(x,gx,t_{0}))>M(x,gx,t_{0}),$ a contradiction.

Consequently $gx=x.$

Again if $fx\neq x,$ $0<M(x,fx,t_{1})<1,$ for some $t_{1}>0.$

Since $X$ is non-Archimedean, $\forall~{}n\in\mathbb{N}$ we have

$M(x,fx,t_{1})\geq M(x,x_{r_{n}+2},t_{1})*M(x_{r_{n}+2},fx,t_{1})\\\
=M(x,x_{r_{n}+2},t_{1})*M(fx,gx_{r_{n}+1},t_{1})\\\ \geq
M(x,x_{r_{n}+1},t_{1})*M(x_{r_{n}+1},x_{r_{n}+2},t_{1})*M(fx,gx_{r_{n}+1},t_{1}).$

Since $\beta(x_{r_{n}},x,t_{1})\leq 1,~{}\forall~{}n\in\mathbb{N}$ we have
$\beta(x,x_{r_{n}+1},t_{1})\leq 1,~{}\forall~{}n\in\mathbb{N}$ by (i).

Thus $\forall~{}n\in\mathbb{N},$

$M(x,fx,t_{1})\geq
M(x,x_{r_{n}+1},t_{1})*M(x_{r_{n}+1},x_{r_{n}+2},t_{1})*(\beta(x,x_{r_{n}+1},t_{1})M(fx,gx_{r_{n}+1},t_{1}))\\\
\geq
M(x,x_{r_{n}+1},t_{1})*M(x_{r_{n}+1},x_{r_{n}+2},t_{1})*\psi(\min\\{M(x,x_{r_{n}+1},t_{1}),M(x,fx,t_{1}),M(x_{r_{n}+1},\\\
gx_{r_{n}+1},t_{1})\\})\\\ \geq
M(x,x_{r_{n}+1},t_{1})*M(x_{r_{n}+1},x_{r_{n}+2},t_{1})*\psi(\min\\{M(x,x_{r_{n}},t_{1})*M(x_{r_{n}},x_{r_{n}+1},t_{1}),M(x,fx,\\\
t_{1}),M(x_{r_{n}+1},gx_{r_{n}+1},t_{1})\\})\\\ \geq
M(x,x_{r_{n}+1},t_{1})*M(x_{r_{n}+1},x_{r_{n}+2},t_{1})*\psi(\min\\{M(x,x_{r_{n}},t_{1})*M(x_{r_{n}},x_{r_{n}+1},t_{1}),M(x,fx,\\\
t_{1}),M(x_{r_{n}+1},x_{r_{n}+2},t_{1})\\})\to\psi(M(x,fx,t_{1}))$ as
$n\to\infty,$ since $*$ is continuous and $(x_{n})$ is $G$-Cauchy.

Thus $M(x,fx,t_{1})\geq\psi(M(x,fx,t_{1}))>M(x,fx,t_{1}),$ a contradiction.

Consequently $fx=x.$

Thus $f,g$ have a common fixed point $x.$

Again, if all the $r_{n}$’s are odd, it can be similarly shown that $fx=gx=x.$

Uniqueness of the fixed point can be proved by proceeding as in Theorem 3.1. ∎

######  Example 3.3.

Consider the fuzzy metric space $(X,M,\cdot)$ of Example 3.1. Let $f,g$ be
self-mappings on $X$ such that

$fx=\begin{cases}1&\text{if }x\in\mathbb{Q}\\\ \vskip
9.0pt\cr\frac{1}{\sqrt{2}}&\text{otherwise}\end{cases}$

and

$gx=\begin{cases}1&\text{if }x=1\\\ \vskip 9.0pt\cr\frac{1}{\sqrt{2}}&\text{if
}x=\frac{1}{\sqrt{2}}\\\ \vskip
9.0pt\cr\frac{1}{2}&\text{otherwise}\end{cases}$

If $\beta:X^{2}\times(0,\infty)\to(0,\infty)$ is defined by

$\beta(x,y,t)=\begin{cases}1&\text{if }x=y=1\\\ \vskip 9.0pt\cr
4&\text{otherwise}\end{cases}$

then $(f,g)$ is a pair of $\beta$-$\psi$-fuzzy contractive mappings,
$\forall~{}\psi\in\Psi.$

It is easy to see that conditions (i)$-$(ii) of Theorem 3.4 hold immediately.

So, in view of Theorem 3.4, $f,g$ have a common fixed point.

However the fixed point is not unique. Indeed $x=1$ and $x=\frac{1}{\sqrt{2}}$
are two such points.

By putting $\beta\equiv 1$ in Theorem 3.4 we see that for a GV fuzzy metric
space, non-Archimedeaness replaces the continuity of $f,g$ in Corollary 3.1:

######  Corollary 3.4.

Let $(X,M,*)$ be a non-Archimedean weak $G$-complete fuzzy metric space and
$f,g$ be self-mappings on $X.$ Suppose for some $\psi\in\Psi,$
$M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),~{}\forall~{}x,y\in
X,t>0.$ Then $f$ and $g$ have a unique common fixed point in $X.$

We will later see that the above conclusion of the above corollary holds, in
fact, in a complete GV fuzzy metric space.

######  Corollary 3.5.

Let $(X,M,*)$ be a non-Archimedean weak $G$-complete fuzzy metric space and
$f$ be a self-mapping on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,f)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,f)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1,$ $\forall~{}t>0.$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and a subsequence $(x_{r_{n}})$ of
$(x_{n})$ with $\lim\limits_{n\to\infty}x_{r_{n}}=x,$ we have
$\beta(x_{r_{n}},x,t)\leq 1,~{}\forall~{}n\in\mathbb{N},t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0,$ then the fixed
point is unique.

In what follows, we show that the additional hypothesis:

for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k_{0}\in\mathbb{N}$ such
that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$ and $t>0$

enables the conclusion of Theorem 3.4 to work in a complete GV fuzzy metric
space.

######  Theorem 3.5.

Let $(X,M,*)$ be a non-Archimedean complete fuzzy metric space and $f,g$ be
self-mappings on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,g)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1,$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and $\lim\limits_{n\to\infty}x_{n}=x,$ we
have $\beta(x_{n},x,t)\leq 1,~{}\forall~{}n\in\mathbb{N},t>0,$

(iv) for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k^{\prime}\in\mathbb{N}$
such that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with
$m\geq n>k^{\prime}$ and $t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0,$ then the fixed
point is unique.

###### Proof.

Proceeding as in Theorem 3.1, we obtain the $G$-Cauchy sequence $(x_{n})$ in
$X$ given by
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$
such that $\beta(x_{n},x_{n+1},t)\leq 1,~{}\forall~{}n=0,1,2,\cdots$ and
$t>0.$ Thus by condition (iv), there exists $k_{1}\in\mathbb{N}$ such that
$\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{1}$ and $t>0.$

We show that $(x_{n})$ is a Cauchy sequence. If not, there exist
$\epsilon\in(0,1),t>0$ and $k_{0}\in\mathbb{N}$ with $k_{0}>k_{1}$ such that
for $k\in\mathbb{N}$ with $k\geq k_{0},$ we have
$m^{\prime}(k),n(k)\in\mathbb{N}$ with $m^{\prime}(k)>n(k)\geq k$ satisfying
$M(x_{m^{\prime}(k)},x_{n(k)},t)\leq 1-\epsilon.$

For each $k\geq k_{0},$ we set $m(k)$ to be the smallest positive integer
exceeding $n(k)$ such that $M(x_{m(k)},x_{n(k)},t)\leq 1-\epsilon\text{ and
}M(x_{m(k)-1},x_{n(k)},t)>1-\epsilon.$

Therefore $\forall~{}k\geq k_{0},$

$1-\epsilon\geq M(x_{m(k)},x_{n(k)},t)\\\ ~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq
M(x_{m(k)-1},x_{n(k)},t)*M(x_{m(k)-1},x_{m(k)},t)\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq(1-\epsilon)*M(x_{m(k)-1},x_{m(k)},t).$

Taking limit as $k\to\infty,$ we obtain
$\lim\limits_{k\to\infty}M(x_{m(k)},x_{n(k)},t)=1-\epsilon.$

Now for each $k\in\mathbb{N}$ with $k\geq k_{0},$ we set

$u(k)=M(x_{m(k)},x_{m(k)+2},t)*M(x_{m(k)+2},x_{n(k)+2},t)*M(x_{n(k)},x_{n(k)+2},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ odd, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+2},t)*M(x_{m(k)+2},x_{n(k)+1},t)*M(x_{n(k)},x_{n(k)+1},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ odd, $n(k)=$ odd}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*M(x_{m(k)+1},x_{n(k)+2},t)*M(x_{n(k)},x_{n(k)+2},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ even, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*M(x_{m(k)+1},x_{n(k)+1},t)*M(x_{n(k)},x_{n(k)+1},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ even, $n(k)=$ odd}$

and

$v(k)=M(x_{m(k)},x_{m(k)+2},t)*\beta(x_{m(k)+1},x_{n(k)+1},t)M(x_{m(k)+2},x_{n(k)+2},t)*M(x_{n(k)},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{n(k)+2},t),\text{ if $m(k)=$ odd, $n(k)=$
even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+2},t)*\beta(x_{m(k)+1},x_{n(k)},t)M(x_{m(k)+2},x_{n(k)+1},t)*M(x_{n(k)},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{n(k)+1},t),\text{ if $m(k)=$ odd, $n(k)=$
odd}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*\beta(x_{m(k)},x_{n(k)+1},t)M(x_{m(k)+1},x_{n(k)+2},t)*M(x_{n(k)},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}x_{n(k)+2},t),\text{ if $m(k)=$ even, $n(k)=$
even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*\beta(x_{m(k)},x_{n(k)},t)M(x_{m(k)+1},x_{n(k)+1},t)*M(x_{n(k)},x_{n(k)+1},\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}t),\text{ if $m(k)=$ even, $n(k)=$ odd}$

Since $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$, we have $\forall~{}k\geq k_{0},$ $1-\epsilon\geq
M(x_{m(k)},x_{n(k)},t)\geq u(k)\geq v(k)\cdots(*).$

Now for each $k\in\mathbb{N}$ with $k\geq k_{0},$ we also set

$w(k)=\psi(\min\\{M(x_{m(k)+1},x_{n(k)+1},t),M(x_{m(k)+1},x_{m(k)+2},t),M(x_{n(k)+1},x_{n(k)+2},t)\\}),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ odd, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=\psi(\min\\{M(x_{m(k)+1},x_{n(k)},t),M(x_{m(k)+1},x_{m(k)+2},t),M(x_{n(k)},x_{n(k)+1},t)\\}),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ odd, $n(k)=$ odd}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=\psi(\min\\{M(x_{m(k)},x_{n(k)+1},t),M(x_{m(k)},x_{m(k)+1},t),M(x_{n(k)+1},x_{n(k)+2},t)\\}),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ even, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=\psi(\min\\{M(x_{m(k)},x_{n(k)},t),M(x_{m(k)},x_{m(k)+1},t),M(x_{n(k)},x_{n(k)+1},t)\\}),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ even, $n(k)=$ odd}$

and

$w^{\prime}(k)=\psi(\min\\{M(x_{m(k)},x_{m(k)+1},t)*M(x_{m(k)},x_{n(k)},t)*M(x_{n(k)},x_{n(k)+1},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}M(x_{m(k)+1},x_{m(k)+2},t),M(x_{n(k)+1},x_{n(k)+2},t)\\}),\text{if
$m(k)=$ odd, $n(k)=$ even}\\\
=\psi(\min\\{M(x_{m(k)},x_{m(k)+1},t)*M(x_{m(k)},x_{n(k)},t),M(x_{m(k)+1},x_{m(k)+2},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}M(x_{n(k)},x_{n(k)+1},t)\\}),\text{if $m(k)=$ odd,
$n(k)=$ odd}\\\
=\psi(\min\\{M(x_{m(k)},x_{n(k)},t)*M(x_{n(k)},x_{n(k)+1},t),M(x_{m(k)},x_{m(k)+1},t),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}M(x_{n(k)+1},x_{n(k)+2},t)\\}),\text{if $m(k)=$
even, $n(k)=$ even}\\\
=\psi(\min\\{M(x_{m(k)},x_{n(k)},t),M(x_{m(k)},x_{m(k)+1},t),M(x_{n(k)},x_{n(k)+1},t)\\}),\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{if $m(k)=$ even, $n(k)=$ odd}.$

For each $k\in\mathbb{N}$ with $k\geq k_{0},$ we further set

$r(k)=M(x_{m(k)},x_{m(k)+2},t)*w(k)*M(x_{n(k)},x_{n(k)+2},t),\text{if $m(k)=$
odd, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+2},t)*w(k)*M(x_{n(k)},x_{n(k)+1},t),\text{if
$m(k)=$ odd, $n(k)=$ odd}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*w(k)*M(x_{n(k)},x_{n(k)+2},t),\text{if
$m(k)=$ even, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*w(k)*M(x_{n(k)},x_{n(k)+1},t),\text{if
$m(k)=$ even, $n(k)=$ odd}$

and

$r^{\prime}(k)=M(x_{m(k)},x_{m(k)+2},t)*w^{\prime}(k)*M(x_{n(k)},x_{n(k)+2},t),\text{if
$m(k)=$ odd, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+2},t)*w^{\prime}(k)*M(x_{n(k)},x_{n(k)+1},t),\text{if
$m(k)=$ odd, $n(k)=$ odd}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*w^{\prime}(k)*M(x_{n(k)},x_{n(k)+2},t),\text{if
$m(k)=$ even, $n(k)=$ even}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}=M(x_{m(k)},x_{m(k)+1},t)*w^{\prime}(k)*M(x_{n(k)},x_{n(k)+1},t),\text{if
$m(k)=$ even, $n(k)=$ odd}.$

Since $\lim\limits_{k\to\infty}w^{\prime}(k)=\psi(1-\epsilon),$ we have
$\lim\limits_{k\to\infty}r^{\prime}(k)=\psi(1-\epsilon).$

Using $(*),$ we have $1-\epsilon\geq M(x_{m(k)},x_{n(k)},t)\geq u(k)\geq
v(k)\geq r(k)\geq r^{\prime}(k),~{}\forall~{}k\geq k_{0}.$ Thus by taking
limit as $k\to\infty,$ we have $1-\epsilon\geq\psi(1-\epsilon)>1-\epsilon,$
which leads to a contradiction.

Thus $(x_{n})$ is a Cauchy sequence and hence converges to some $x\in X.$

We now show that $fx=gx=x.$

If $gx\neq x,$ then there exists $t_{0}>0$ such that $0<M(x,gx,t_{0})<1.$

Since $M(x,gx,t_{0})\geq M(x,x_{2n+1},t_{0})*M(fx_{2n},gx,t_{0})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq
M(x,x_{2n+1},t_{0})*\beta(x_{2n},x,t_{0})M(fx_{2n},gx,t_{0})\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}\geq
M(x,x_{2n+1},t_{0})*\psi(\min\\{M(x,x_{2n},t_{0}),M(x,gx,t_{0}),M(x_{2n+1},x_{2n},t_{0})\\})$

so by taking limit as $n\to\infty$ we have
$M(x,gx,t_{0})\geq\psi(M(x,gx,t_{0})>M(x,gx,t_{0}),$ which leads to a
contradiction.

Thus $gx=x.$ Similarly, $fx=x.$ Hence $fx=gx=x.$

Uniqueness of the fixed point can be proved by proceeding as in Theorem 3.1. ∎

######  Corollary 3.6.

Let $(X,M,*)$ be a non-Archimedean complete fuzzy metric space and $f$ be a
self-mapping on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $(f,f)$ is a pair of
$\beta$-$\psi$-fuzzy contractive mappings such that

(i) $(f,f)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1,$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and $\lim\limits_{n\to\infty}x_{n}=x,$ we
have $\beta(x_{n},x,t)\leq 1,~{}\forall~{}n\in\mathbb{N},t>0,$

(iv) for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k_{0}\in\mathbb{N}$ such
that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$ and $t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0,$ then the fixed
point is unique.

By putting $\beta\equiv 1,$ in Theorem 3.5 we have the following:

######  Corollary 3.7.

Let $(X,M,*)$ be a non-Archimedean complete fuzzy metric space and $f,g$ be
self-mappings on $X.$ Suppose for some $\psi\in\Psi,$

$M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),~{}\forall~{}x,y\in
X,t>0.$

Then $f$ and $g$ have a unique common fixed point in $X.$

We encourage the readers to compare the above result with Corollary 3.1 and
Theorem 3.2.

######  Theorem 3.6.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
continuous self-mappings on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(gfx,fgy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},gfx_{0},t),M(gfx_{0},fgfx_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Define a sequence $(x_{n})$ as follows:
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$.

Then proceeding as Theorem 3.1, we obtain

$\beta(x_{n},x_{n+1},t),\beta(x_{n+1},x_{n},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0.$

Let $n$ be a positive integer bigger than $1$.

If $n\text{ is }\text{odd},$ then for chosen $t>0,$
$M(x_{n-1},x_{n},t)>0\implies$

$M(x_{n+1},x_{n+2},t)=M(gfx_{n-1},fgx_{n},t)$

$\geq\beta(x_{n-1},x_{n},t)M(gfx_{n-1},fgx_{n},t)$

$\geq\psi(\min\\{M(x_{n-1},x_{n},t),M(x_{n-1},fx_{n-1},t),M(x_{n},gx_{n},t)\\})$

$=\psi(\min\\{M(x_{n-1},x_{n},t),M(x_{n},x_{n+1},t)\\})$

If $n$ is even, then for chosen $t>0,$ $M(x_{n-1},x_{n},t)>0\implies$

$M(x_{n+1},x_{n+2},t)=M(gfx_{n},fgx_{n-1},t)$

$\geq\beta(x_{n},x_{n-1},t)M(gfx_{n},fgx_{n-1},t)$

$\geq\psi(\min\\{M(x_{n},x_{n-1},t),M(x_{n},fx_{n},t),M(x_{n-1},gx_{n-1},t)\\})$

$=\psi(\min\\{M(x_{n-1},x_{n},t),M(x_{n},x_{n+1},t)\\}).$

Consequently, $\forall~{}n\geq 2,t>0,$ $M(x_{n-1},x_{n},t)>0\implies$
$\text{either }M(x_{n+1},x_{n+2},t)$ $\geq\psi(M(x_{n-1},x_{n},t)\text{ or
}M(x_{n+1},x_{n+2},t)\geq\psi(M(x_{n},x_{n+1},t)).$

Since $M(x_{1},x_{2},t),M(x_{2},x_{3},t)>0,$ we conclude
$M(x_{n-1},x_{n},t)>0,~{}\forall~{}n\geq 2,t>0.$

Let us set
$S_{t}=\min\\{M(x_{1},x_{2},t),M(x_{2},x_{3},t)\\},~{}\forall~{}t>0.$

Then $M(x_{2n-1},x_{2n},t)\geq\psi^{n-1}(S_{t})$ and
$M(x_{2n},x_{2n+1},t)\geq\psi^{n-1}(S_{t}),~{}\forall~{}n\geq 2,t>0.$

Now
$\forall~{}t>0,S_{t}>0\implies\lim\limits_{n\to\infty}\psi^{n-1}(S_{t})=1.$

Thus $\forall~{}t>0,$
$\lim\limits_{n\to\infty}M(x_{2n-1},x_{2n},t)=\lim\limits_{n\to\infty}M(x_{2n},x_{2n+1},t)=1,$
and consequently, $\lim\limits_{n\to\infty}M(x_{n},x_{n+1},t)=1.$

So $(x_{n})$ is a $G$-Cauchy sequence in $X.$

Now proceeding as Theorem 3.1, we conclude that $x$ is a common fixed point of
$f$ and $g$ which is unique if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$
and for $x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0.$

Hence the result follows. ∎

######  Example 3.4.

We note that Example 3.1 satisfies the hypothesis of Theorem 3.6.

Let us now replace the value of $\beta,$ in Example 3.1, with $1.$ It is clear
that the modified example satisfies the hypothesis of Theorem 3.6 as well.

We now show that the modified example does not satisfy the hypothesis of
Theorem 3.1.

Suppose otherwise. Then by putting $x=\frac{1}{2},y=1,t=1$ in

$\beta(x,y,t)M(fx,gy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

we obtain
$M\left(\frac{1}{2},1,1\right)\geq\psi\left(M\left(\frac{1}{2},1,1\right)\right),$
which is a contradiction since $\psi(x)>x,~{}\forall~{}x\in(0,1).$

######  Remark 3.1.

We note that Example 3.2 satisfies the hypothesis of first part of Theorem
3.6. Since $f,g$ in the said Example have more than one common fixed points,
the last condition is essential to ensure the uniqueness of the common fixed
point in Theorem 3.6.

######  Corollary 3.8.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f$ be a
continuous self-mapping on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(f^{2}x,f^{2}y,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,fy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,f)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},f^{2}x_{0},t),M(f^{2}x_{0},f^{3}x_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

######  Corollary 3.9.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f$ be a
continuous self-mapping on $X.$ Suppose for some $\psi\in\Psi,$ and a mapping
$\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(fx,fy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t)\\},$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,I)$ is a symmetric pair of $\beta$-admissible mappings ($I$ being the
identity self-mapping on $X),$

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},f^{2}x_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Immediate by setting $g=I$ in Theorem 3.6. ∎

In the next theorem, we omit the continuity hypothesis of one of $f,g$ in
Theorem 3.6.

######  Theorem 3.7.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
self-mappings on $X$ such that one of them is continuous. Suppose for some
$\psi\in\Psi,$ and a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$
$M(x,y,t)>0\implies$

$\beta(x,y,t)M(gfx,fgy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},gfx_{0},t),M(gfx_{0},fgfx_{0},t)>0,$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and a subsequence $(x_{r_{n}})$ of
$(x_{n})$ with $\lim\limits_{n\to\infty}x_{r_{n}}=x,$ we have
$\beta(x_{r_{n}},x,t),\beta(x,x_{r_{n}},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Proceeding as in Theorem 3.6, we obtain the $G$-Cauchy sequence $(x_{n})$ in
$X$ given by
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$
such that $\beta(x_{n},x_{n+1},t),\beta(x_{n+1},x_{n},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0.$

Since $(X,M,*)$ is weak $G$-complete, there exists a subsequence $(x_{r_{n}})$
of $(x_{n})$ such that $(x_{r_{n}})$ converges to some $x\in X$ where
$r_{n}$’s are either all even or all odd.

Without loss of generality, suppose all the $r_{n}$’s are even.

Case I: Let $f$ be continuous. Then $\forall~{}t>0,n\in\mathbb{N},$

$M(x,fx,t)\geq
M\left(x,x_{r_{n}+1},\frac{t}{2}\right)*M\left(x_{r_{n}+1},fx,\frac{t}{2}\right)\\\
=M\left(x,x_{r_{n}+1},\frac{t}{2}\right)*M\left(fx_{r_{n}},fx,\frac{t}{2}\right)\\\
=M\left(x,x_{r_{n}},\frac{t}{4}\right)*M\left(x_{r_{n}},x_{r_{n}+1},\frac{t}{4}\right)*M\left(fx_{r_{n}},fx,\frac{t}{2}\right)$

Taking limit as $n\to\infty,$ we see that the right hand side tends to $1.$

Thus we have $M(x,fx,t)=1,~{}\forall~{}t>0$ and consequently, $fx=x.$

We now show that $gx=x.$ Choose $t_{1}>0.$

Then for sufficiently large values of $n,$ $M(x,gx,t_{1})\\\ \geq
M(x,x_{r_{n}},\frac{t_{1}}{5})*M(x_{r_{n}},x_{r_{n}+1},\frac{t_{1}}{5})*M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5})*M(x_{r_{n}+2},x_{r_{n}+3},\frac{t_{1}}{5})*M(x_{r_{n}+3},\\\
gx,\frac{t_{1}}{5})\\\ \geq
M(x,x_{r_{n}},\frac{t_{1}}{5})*M(x_{r_{n}},x_{r_{n}+1},\frac{t_{1}}{5})*M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5})*M(x_{r_{n}+2},x_{r_{n}+3},\frac{t_{1}}{5})*M(fgx_{r_{n}+1},\\\
gfx,\frac{t_{1}}{5})\\\ \geq
M(x,x_{r_{n}},\frac{t_{1}}{5})*M(x_{r_{n}},x_{r_{n}+1},\frac{t_{1}}{5})*M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5})*M(x_{r_{n}+2},x_{r_{n}+3},\frac{t_{1}}{5})*(\beta(x,x_{r_{n}+1},\\\
\frac{t_{1}}{5})M(gfx,fgx_{r_{n}+1},\frac{t_{1}}{5}))\text{
(since}\lim\limits_{n\to\infty}x_{r_{n}+1}=x,~{}\beta(x,x_{r_{n}+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0)\\\ \geq
M(x,x_{r_{n}},\frac{t_{1}}{5})*M(x_{r_{n}},x_{r_{n}+1},\frac{t_{1}}{5})*M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5})*M(x_{r_{n}+2},x_{r_{n}+3},\frac{t_{1}}{5})*\\\
\psi(\min\\{M(x_{r_{n}+1},x,\frac{t_{1}}{5}),M(x_{r_{n}+1},gx_{r_{n}+1},\frac{t_{1}}{5}),M(x,fx,\frac{t_{1}}{5})\\})\\\
\geq
M(x,x_{r_{n}},\frac{t_{1}}{5})*M(x_{r_{n}},x_{r_{n}+1},\frac{t_{1}}{5})*M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5})*M(x_{r_{n}+2},x_{r_{n}+3},\frac{t_{1}}{5})*\\\
\psi(\min\\{M(x_{r_{n}+1},x,\frac{t_{1}}{5}),M(x_{r_{n}+1},x_{r_{n}+2},\frac{t_{1}}{5}),M(x,fx,\frac{t_{1}}{5})\\}.$

Taking limit as $n\to\infty,$ we see that the right hand side tends to $1.$

Thus $M(x,gx,t_{1})=1,~{}\forall~{}t_{1}>0\implies x=gx.$

Case II: Let $g$ be continuous. Then we can similiarly show that $x$ is a
common fixed point of $f,g.$

Again, if all the $r_{n}$’s are odd, it can be similarly shown that $fx=gx=x.$

Uniqueness of the fixed point can be proved by proceeding as in Theorem 3.1. ∎

By putting $\beta\equiv 1$ in Theorem 3.7 we have the following:

######  Corollary 3.10.

Let $(X,M,*)$ be a weak $G$-complete (KM) fuzzy metric space and $f,g$ be
self-mappings on $X$ such that one of them is continuous. Suppose for some
$\psi\in\Psi,$ $M(x,y,t)>0\implies$

$M(gfx,fgy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that
$M(fx_{0},gfx_{0},t),M(gfx_{0},fgfx_{0},t)>0,$ $\forall~{}t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if for $x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then
the fixed point is unique.

We encourage the readers to compare the last result with Corollary 3.1 and
Theorem 3.2.

######  Example 3.5.

Consider the fuzzy metric space $(X,M,\cdot)$ of Example 3.1. Let $f,g$ be
self-mappings on $X$ such that

$fx=\begin{cases}1&\text{if }x\in\mathbb{Q}\\\ \vskip
9.0pt\cr\frac{1}{2}&\text{otherwise}\end{cases}.$

and $gx=1,~{}\forall~{}x\in X.$

Clearly, $g$ is continuous on $X$ but $f$ is not.

Since $gfx=fgy=1,~{}\forall~{}x,y\in X$ we have

$M(gfx,fgy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

$\forall~{}x,y\in X,t>0$ and $\psi\in\Psi.$

Thus by setting $\beta\equiv 1$ we see that $f,g$ satisfy the hypothesis of
Theorem 3.7.

Here $1$ is a common fixed point of $f$ and $g$ which is clearly unique.

The following question is open:

######  Question 1.

Can the continuity hypothesis of both $f$ and $g$ in Theorem 3.7 be omitted?

In what follows, we show that the additional hypothesis:

for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k_{0}\in\mathbb{N}$ such
that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$ and $t>0$

enables the conclusion of Theorem 3.7 to work in a complete (KM) fuzzy metric
space.

######  Theorem 3.8.

Let $(X,M,*)$ be a non-Archimedean complete (KM) fuzzy metric space and $f,g$
be self-mappings on $X$ such that one of them is continuous. Suppose for some
$\psi\in\Psi,$ and a mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$
$M(x,y,t)>0\implies$

$\beta(x,y,t)M(gfx,fgy,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,gy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,g)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},gfx_{0},t),M(gfx_{0},fgfx_{0},t)>0,$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and a subsequence $(x_{r_{n}})$ of
$(x_{n})$ with $\lim\limits_{n\to\infty}x_{r_{n}}=x,$ we have
$\beta(x_{r_{n}},x,t),\beta(x,x_{r_{n}},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0,$

(iv) for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k_{0}\in\mathbb{N}$ such
that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$ and $t>0.$

Then $f$ and $g$ have a common fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

###### Proof.

Proceeding as in Theorem 3.6, we obtain the $G$-Cauchy sequence $(x_{n})$ in
$X$ given by
$x_{1}=fx_{0},x_{2}=gx_{1},\cdots,x_{2n+1}=fx_{2n},x_{2n+2}=gx_{2n+1},\cdots$
that will turn out to be Cauchy following an argument similar to Theorem 3.5.
Since $X$ is complete, there exists $x\in X$ such that
$\lim\limits_{n\to\infty}x_{n}=x.$ Similar to Theorem 3.7, we can show that
$fx=gx=x.$ Uniqueness of the common fixed point can be proved by proceeding as
in Theorem 3.1. ∎

######  Corollary 3.11.

Let $(X,M,*)$ be a non-Archimedean complete (KM) fuzzy metric space and $f$ be
a continuous self-mapping on $X$. Suppose for some $\psi\in\Psi,$ and a
mapping $\beta:X^{2}\times(0,\infty)\to(0,\infty),$ $M(x,y,t)>0\implies$

$\beta(x,y,t)M(f^{2}x,f^{2}y,t)\geq\psi(\min\\{M(x,y,t),M(x,fx,t),M(y,fy,t)\\}),$

$\forall~{}x,y\in X,t>0$ such that

(i) $(f,f)$ is a symmetric pair of $\beta$-admissible mappings,

(ii) for some $x_{0}\in X,$ $\beta(x_{0},fx_{0},t)\leq 1$ and
$M(fx_{0},f^{2}x_{0},t),M(f^{2}x_{0},f^{3}x_{0},t)>0,$ $\forall~{}t>0,$

(iii) for each sequence $(x_{n})$ in $X$ with $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$ and a subsequence $(x_{r_{n}})$ of
$(x_{n})$ with $\lim\limits_{n\to\infty}x_{r_{n}}=x,$ we have
$\beta(x_{r_{n}},x,t),\beta(x,x_{r_{n}},t)\leq
1,~{}\forall~{}n\in\mathbb{N},t>0$,

(iv) for each sequence $(x_{n})$ in $X$ satisfying $\beta(x_{n},x_{n+1},t)\leq
1,~{}\forall~{}n\in\mathbb{N},~{}t>0,$ there exists $k_{0}\in\mathbb{N}$ such
that $\beta(x_{m},x_{n},t)\leq 1,~{}\forall~{}m,n\in\mathbb{N}$ with $m\geq
n>k_{0}$ and $t>0.$

Then $f$ has a fixed point in $X.$

Moreover, if $\beta(x,y,t)\leq 1,~{}\forall~{}x,y\in X,t>0$ and for
$x,y~{}(x\neq y)\in X,$ $M(x,y,t)>0,~{}\forall~{}t>0,$ then the fixed point is
unique.

## References

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  * [2] Adhya, S., & Deb Ray, A. (2021). Some properties of Lebesgue fuzzy metric spaces, Sahand Communications in Mathematical Analysis, 18(1), 1-14.
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  * [13] Sherwood, H. (1966). On the completion of probabilistic metric spaces. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 6(1), 62-64.
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  * [15] Turkoglu, D., & Sangurlu, M. (2014). Fixed point theorems for fuzzy $\psi$–contractive mappings in fuzzy metric spaces. Journal of Intelligent & Fuzzy Systems, 26(1), 137-142.
  * [16] Vetro, C. (2011). Fixed points in weak non-Archimedean fuzzy metric spaces. Fuzzy Sets and Systems, 162(1), 84-90.


# Bounds for Rainbow-uncommon Graphs

Blake Bates  Zhanar Berikkyzy  Nick Chiem  Gabriel Elvin  Risa Fines  Maja Lie
Hanna Mikulás  Isaac Reiter  Kevin Zhou

(March 6, 2024)

###### Abstract

We say a graph $H$ is $r$-rainbow-uncommon if the maximum number of rainbow
copies of $H$ under an $r$-coloring of $E(K_{n})$ is asymptotically (as
$n\to\infty$) greater than what is expected from uniformly random
$r$-colorings. Via explicit constructions, we show that for
$H\in\\{K_{3},K_{4},K_{5}\\}$, $H$ is $r$-rainbow-uncommon for all
$r\geq{|V(H)|\choose 2}$. We also construct colorings to show that for $t\geq
6$, $K_{t}$ is $r$-rainbow-uncommon for sufficiently large $r$.

##  1 Background

A classic starting point in Ramsey theory is the following question: if any
two people can identify as either strangers or acquaintances, what is the
minimum number of people needed to guarantee that there exists a set of three
people who are all either complete strangers or mutual acquaintances? Reframed
in the language of graph theory, we can let vertices denote people and draw a
blue edge between two people if they are strangers and a red edge if they are
acquaintances. Then the question is equivalent to: what is the smallest $n$
such that every coloring of the edges of a complete graph $K_{n}$ contains a
monochromatic triangle, i.e. a copy of $K_{3}$ whose edges are either all blue
or all red? The answer to this question is $n=6$, and in 1959, Goodman asked
and answered a more general follow-up question: for an arbitrary $n$, what is
the minimum number of monochromatic triangles over all 2-colorings $E(K_{n})$
[3]. He found an explicit formula, but a consequence of his result is that the
minimum proportion of monochromatic triangles, that is

$\frac{\text{number of monochromatic $K_{3}$}}{\text{total number of
$K_{3}$}},$

approaches $1/4$ as $n\to\infty$. This is perhaps surprising, since $1/4$ is
the expected proportion in a uniformly random coloring, i.e. the minimum can
be achieved, at least asymptotically, by coloring completely randomly. This
led to a new vein of Ramsey theory, classifying graphs as either common or
uncommon: common if the asymptotic minimum can be achieved by uniformly random
colorings, and uncommon if better colorings can be found. See [1, Section 2.6]
for further reading in this area.

This question can be “flipped”: given a fixed graph $H$ and $r$ colors, what
is the best way to color the edges of $K_{n}$ ($n$ large) in order to maximize
the number of rainbow copies of $H$ in $K_{n}$? By rainbow, we mean that each
edge of $H$ is assigned a different color from all the others. Some progress
has been made in this area, and our work builds on the work in [4]. Next, we
paraphrase those authors’ definitions, slightly altered to suit our needs.

In this paper all graphs are simple. If $H$ and $G$ are graphs, a copy of $H$
in $G$ is a subgraph of $G$ that is isomorphic to $H$. We define an
$r$-coloring of a set $X$ to be a surjective function $c:X\to\\{1,\dots,r\\}$.
For any graph $G$, let $E(G)$ denote the set of edges of $G$. Given a graph
$G$ and subgraph $H$, we say $H$ is rainbow if, under a coloring
$c:E(G)\to\\{1,\dots,r\\}$, $c(uv)\neq c(u^{\prime}v^{\prime})$ for all
distinct $uv,u^{\prime}v^{\prime}\in E(H)$.

Results in this area are asymptotic in nature, so we clarify some notation
used throughout. Let $f,g$ be functions of $n$. If there exist constants $C,N$
such that $|f(n)|\leq Cg(n)$ for all $n\geq N$, we write $f=O(g)$.

For the remainder of this section, let $H$ be a fixed graph on $t$ vertices
with $e$ edges. We denote by $M_{\text{rb}}(H;n,r)$ the maximum number of
rainbow copies of $H$ over all $r$-colorings of $E(K_{n})$. Furthermore, we
define

$\displaystyle m_{\text{rb}}(H;n,r)$
$\displaystyle\coloneqq\frac{M_{\text{rb}}(H;n,r)}{\text{total number of
copies of $H$ in $K_{n}$}},$ (1) $\displaystyle m_{\text{rb}}(H;r)$
$\displaystyle\coloneqq\lim_{n\to\infty}m_{\text{rb}}(H;n,r).$ (2)

For a given graph $H$, $m_{\text{rb}}(H;r)$ is known as the $r$-rainbow Ramsey
multiplicity constant. Note that it always exists due to the fact that the
sequence $m_{\text{rb}}(H;n,r)$ is bounded and monotone in $n$, as shown in
[4]. The overarching classification question is: for which graphs $H$ and
number of colors $r$ can $m_{\text{rb}}(H;r)$ be achieved by uniformly random
colorings? Note that, for any fixed copy of $H$ in a uniformly random
$r$-coloring of $E(K_{n})$, there are $r\choose{e}$ distinct colors to choose
that could make the copy of $H$ rainbow, and once those colors are chosen
there are $e!$ ways the subgraph could be colored. Since there are $r^{e}$
possible colorings of this subgraph, the expected proportion of rainbow $H$ in
$G$ is ${r\choose{e}}e!/r^{e}$. This motivates the following definition: a
graph $H$ is $r$-rainbow-common if

$m_{\text{rb}}(H;r)=\frac{{r\choose{e}}e!}{r^{e}}.$ (3)

Otherwise, $H$ is called $r$-rainbow-uncommon.

In this paper, we investigate rainbow-uncommonness of complete graphs,
building upon the following theorem.

######  Theorem 1 ([2, 4]).

$K_{t}$ is ${t\choose{2}}$-rainbow-uncommon for all $t\geq 3$.

The case $t=3$ was proven by Erdős and Hajnal in 1972 [2], and De Silva et al.
showed it for all $t\geq 4$ in 2019 [4]. Our first result extends that of
Erdős and Hajnal and the first two cases from De Silva et al.

######  Theorem 2.

$K_{t}$ is $r$-rainbow-uncommon for all $r\geq{t\choose{2}}$, when $t=3,4,5$.

We also show that for $t\geq 6$, $K_{t}$ is $r$-rainbow-uncommon for all
sufficiently large $r$, stated precisely below.

######  Theorem 3.

Let $t\geq 6$. There exists $r_{t}$ such that $K_{t}$ is $r$-rainbow-uncommon
for all $r\geq r_{t}$.

In the next section, we prove Theorem 3. Theorem 2 is an immediate consequence
of the work done in the proof of Theorem 3, which we show at the end of the
section.

##  2 Main Result

In general, to show $r$-rainbow-uncommonness of a graph $H$, we can find an
$r$-coloring of $E(K_{n})$ with asymptotically strictly more rainbow copies of
$H$ than the number we would expect from a uniformly random coloring:

$\frac{{n\choose{t}}t!}{|\text{Aut($H$)}|}\cdot\frac{{r\choose{e}}e!}{r^{e}}=\frac{{r\choose{e}}e!}{|\text{Aut}(H)|r^{e}}n^{t}+O(n^{t-1})$
(4)

(the number of $H$ in $K_{n}$ times the probability that any given $H$ is
rainbow).

In our proofs, we use this fact and find explicit colorings building on the
iterated blow-up method in De Silva et al. [4]. The main idea is to find a
coloring of $K_{b}$ which contains many rainbow copies of $H$. Then we “blow-
up” the $K_{b}$ by expanding each vertex into a copy of $K_{b}$, and between
each copy make all edges the same color as the original edge between the
original two vertices. We repeat this $k-1$ times to get a coloring of
$K_{n}$, where $n=b^{k}$.

Figure 1: Iterated blowup of a $K_{4}$. The “thick” edges indicate that all
possible edges between the sets of vertices are given the specified color.

An example of an iterated blowup of a $K_{4}$ is given in Figure 1. Note that
this generates a coloring for only a subsequence of $K_{n}$s. However, since
$m_{\text{rb}}(H;n,r)$ is bounded and monotone in $n$ [4], the limit
$m_{\text{rb}}(H;r)$ must exist and every subsequence must converge to that
limit.

The inequalities in the following lemma were used in [4], and we include their
proofs here for completeness.

######  Lemma 1.

Let $t\leq b$, let $a$ denote the number of rainbow copies of $H$ in an
$r$-coloring of $E(K_{b})$, and let $F(n)$ denote the number of rainbow copies
of $H$ in the $K_{n}$ generated from an iterated blow-up of $K_{b}$. Then

$F(n)\geq bF(n/b)+a(n/b)^{t}.$ (5)

Furthermore, the solution to this recurrence is

$F(n)\geq\frac{a}{b^{t}-b}n^{t}+O(n^{t-1}).$ (6)

###### Proof.

First, we derive the inequality (5). Each vertex of $K_{b}$ that we blow up
contains $F(n/b)$ rainbow copies of $H$, contributing the $bF(n/b)$ term.
Additionally, one vertex from each of the $b$ parts can be chosen in
$(n/b)^{t}$ ways, and each choice contains $a$ rainbow copies of $H$.
Therefore, $F(n)\geq bF(n/b)+a(n/b)^{t}$. For an example of this bound, see
Figure 1. We solve this recurrence using generating functions. First, we
transform the recurrence into something that is easier to solve by making the
substitution $n=b^{k}$, where $k$ is the number of iterations in the blow-up.
We define the function $f(k)=F(b^{k})$ to simplify (5) and get the following
recurrence:

$f(k)\geq bf\left(k-1\right)+a\left(b^{k-1}\right)^{t}.$

Let $A(x)=\sum_{k=0}^{\infty}f(k)x^{k}$ be the generating function. Since
$f(0)=0$, we have:

$\displaystyle A(x)$ $\displaystyle=\sum_{k=1}^{\infty}f(k)x^{k}$
$\displaystyle\geq\sum_{k=1}^{\infty}\left[bf(k-1)+a\left(b^{k-1}\right)^{t}\right]x^{k}$
$\displaystyle=b\sum_{k=1}^{\infty}f(k-1)x^{k}+a\sum_{k=1}^{\infty}(b^{k-1})^{t}x^{k}$
$\displaystyle=bx\sum_{k=1}^{\infty}f(k-1)x^{k-1}+\frac{a}{b^{t}}\sum_{k=1}^{\infty}(b^{t}x)^{k}$
$\displaystyle=bxA(x)+\frac{a}{b^{t}}\left[\sum_{k=0}^{\infty}(b^{t}x)^{k}-1\right]$
$\displaystyle=bxA(x)+\frac{a}{b^{t}}\left[\frac{1}{1-b^{t}x}-1\right]$
$\displaystyle=bxA(x)+\frac{ax}{1-b^{t}x}.$

Note that the geometric series $\sum(b^{t}x)^{k}$ converges for $|x|<b^{-t}$,
which also ensures $1-bx>0$. Therefore, isolating $A(x)$ we get

$A(x)\geq\frac{ax}{(1-bx)(1-b^{t}x)}.$

Next, using partial fraction decomposition, we can rewrite $A(x)$ as a power
series and obtain the closed form of the recurrence. For real numbers $B$ and
$C$, if we set

$\frac{ax}{(1-bx)(1-b^{t}x)}=\frac{B}{1-bx}+\frac{C}{1-b^{t}x},$

we obtain $B=\frac{a}{b-b^{t}}$ and $C=-\frac{a}{b-b^{t}}$. Therefore:

$\displaystyle A(x)$
$\displaystyle\geq\frac{a}{b-b^{t}}\cdot\frac{1}{1-bx}-\frac{a}{b-b^{t}}\cdot\frac{1}{1-b^{t}x}$
$\displaystyle=\frac{a}{b-b^{t}}\sum_{k=0}^{\infty}(bx)^{k}-\frac{a}{b-b^{t}}\sum_{k=0}^{\infty}(b^{t}x)^{k}$
$\displaystyle=\sum_{k=0}^{\infty}\left[\frac{a\left(b^{tk}-b^{k}\right)}{b^{t}-b}\right]x^{k}.$

Therefore, $f(k)\geq\frac{a\left(b^{tk}-b^{k}\right)}{b^{t}-b}$. Substituting
$n=b^{k}$, we get that

$F(n)\geq\frac{a(n^{t}-n)}{b^{t}-b}=\frac{a}{b^{t}-b}n^{t}+O(n^{t-1}),$

as desired. ∎

Combining (4) with (6), it suffices to find a coloring of some $K_{b}$ with
$a$ rainbow copies of $H$, where

$a>\frac{b(b^{t-1}-1){r\choose{e}}e!}{|\text{Aut}(H)|r^{e}}.$ (7)

Before stating and proving the main result, we give an explicit example that
serves as a foundation and motivation for some of the techniques used in the
proof. Note that the following was proved for $r=3$ by Erdős and Hajnal in
1972 [2].

######  Proposition 1.

$K_{3}$ is $r$-rainbow-uncommon for all $r\geq 3$.

###### Proof.

We will define a coloring of $E(K_{r})$ in which every triangle is rainbow and
then use the iterated blowup method to determine a lower bound of the maximum
number of rainbow triangles the $K_{n}$ can contain. For such coloring of
$E(K_{r})$, each color class must form a matching. We call it a parallel
$r$-coloring. Examples with $r=7$ and $r=8$ are provided in Figure 2.

012345601234567 Figure 2: Parallel $r$-coloring defined in the proof of
Proposition 1. In particular, there can be no non-rainbow triangles.

Such coloring can be constructed for all $r\geq 3$, and it is convenient to
describe the parallel $r$-coloring for $r$ odd and even separately. Let $c$
denote the $r$-coloring of $E(K_{r})$ with vertices $\\{0,1,\ldots,r-1\\}$.

  * •

when $r$ is odd, for each $k=1,\ldots,r$ color $c(\\{k+i\;(\text{mod
}r),k-i\;(\text{mod }r)\\})=k$ for all $i=1,\ldots,\frac{r-1}{2}$,

  * •

when $r$ is even, for each $k=1,\ldots,\frac{r}{2}$ color
$c(\\{k+i\;(\text{mod }r),k-i\;(\text{mod }r)\\})=k$ for all
$i=1,\ldots,\frac{r}{2}-1$ and for each $k=\frac{r}{2}+1,\ldots,r$ color
$c(\\{k+i\;(\text{mod }r),k-i+1\;(\text{mod }r)\\})=k$ for all
$i=1,\ldots,\frac{r}{2}$.

It follows that no vertex has two incident edges of the same color, and hence
every $K_{3}$ must be rainbow. By the proof of Lemma 1, the number of rainbow
copies of $K_{3}$ in a $K_{n}$ generated by the iterated blow-up of $K_{r}$
with coloring $c$ is at least $\frac{r-2}{6(r+1)}(n^{3}-n)$, while the
expected number of rainbow copies in a uniformly random coloring is
$\frac{(r-1)(r-2)}{6r^{2}}n(n-1)(n-2)$. It is straightforward to check that
this construction satisfies the inequality (7). ∎

The next two results give us an upper bound the number of non-rainbow $K_{t}$s
in $K_{r}$ under a parallel $r$-coloring.

######  Lemma 2.

Let $r\geq 6$, and let $c:E(K_{r})\rightarrow\\{1,\dots,r\\}$ be a parallel
$r$-coloring. Then there are at most $r\binom{\lfloor r/2\rfloor}{2}$ non-
rainbow copies of $K_{4}$ in $K_{r}$ under $c$.

###### Proof.

Given $r\geq 6$, let $c$ be a parallel $r$-coloring of $E(K_{r})$. Notice that
under this coloring, every non-rainbow $K_{4}$ in $K_{r}$ uses exactly two
edges from some color class. Therefore, for each color, there are at most
$\binom{\lfloor r/2\rfloor}{2}$ non-rainbow copies of $K_{4}$ that have that
color repeated. Thus, since there are $r$ colors in total, the number of non-
rainbow $K_{4}$s in $K_{r}$ is at most $r\binom{\lfloor r/2\rfloor}{2}$. ∎

######  Lemma 3.

Let $t\geq 4$, and let $r\geq{{t}\choose{2}}$ and
$c:E(K_{r})\rightarrow\\{1,\dots,r\\}$ be a parallel $r$-coloring. Then there
are at most $r\binom{\lfloor r/2\rfloor}{2}\binom{r-4}{t-4}$ non-rainbow
copies of $K_{t}$ in $K_{r}$ under $c$.

###### Proof.

Given $t\geq 4$ and $r\geq{{t}\choose{2}}$, let $c$ be a parallel $r$-coloring
of $E(K_{r})$. Notice that every non-rainbow $K_{t}$ in $K_{r}$ contains a
non-rainbow $K_{4}$. By Lemma 2, there are at most $r\binom{\lfloor
r/2\rfloor}{2}$ non-rainbow $K_{4}$s in $K_{r}$ under the parallel
$r$-coloring, and each non-rainbow $K_{4}$ is contained in at most
$\binom{r-4}{t-4}$ $K_{t}$s because we may select the remaining $t-4$ vertices
of $K_{t}$ from $K_{r}$ in $\binom{r-4}{t-4}$ ways. Therefore, the number of
non-rainbow copies of $K_{t}$ in $K_{r}$ is at most $r\binom{\lfloor
r/2\rfloor}{2}\binom{r-4}{t-4}$ under $c$. ∎

We are now ready to prove our main result, restated (and adjusted slightly)
below.

######  Theorem 4.

Let $t\geq 4$. There exists $r_{t}\geq{t\choose{2}}$ such that for all $r\geq
r_{t}$, $K_{t}$ is $r$-rainbow-uncommon.

###### Proof.

Let $t\geq 4$ and $r\geq{{t}\choose{2}}$, and let $c$ be a parallel
$r$-coloring of $E(K_{r})$. By Lemma 3, there are at most $r\binom{\lfloor
r/2\rfloor}{2}\binom{r-4}{t-4}$ non-rainbow copies of $K_{t}$ in $K_{r}$ under
$c$. Therefore, there are at least

$\binom{r}{t}-r\binom{\lfloor
r/2\rfloor}{2}\binom{r-4}{t-4}\geq\binom{r}{t}\left(1-\frac{r\cdot
t!}{8(t-4)!(r-1)(r-3)}\right)$

rainbow copies of $K_{t}$ in $K_{r}$. In the inequality above, note that
$\binom{\lfloor r/2\rfloor}{2}\leq\frac{r(r-2)}{8}$.

We will use the iterated blow-up method described at the beginning of this
section with parameters $H=K_{t}$, $b=r$, and
$a=\binom{r}{t}\left(1-\frac{r\cdot t!}{8(t-4)!(r-1)(r-3)}\right)$. Note that
this implies $|Aut(H)|=t!$ and $e=\binom{t}{2}$. Therefore, it suffices to
show that the inequality (7) is satisfied, i.e. we will show that

$\binom{r}{t}\left(1-\frac{r\cdot
t!}{8(t-4)!(r-1)(r-3)}\right)>\frac{r(r^{t-1}-1){r\choose{\binom{t}{2}}}\binom{t}{2}!}{t!r^{\binom{t}{2}}}.$
(8)

Rearranging the terms in the inequality (8), one can show that it is
equivalent to

$\left(1-\frac{r\cdot
t!}{8(t-4)!(r-1)(r-3)}\right)r^{t(t-1)/2}>\frac{(r-t)!}{(r-\binom{t}{2})!}(r^{t}-r).$

Notice that the term $\frac{(r-t)!}{(r-\binom{t}{2})!}$ on the right hand side
of the inequality above can be written as

$(r-t)(r-t-1)\cdots\left(r-\binom{t}{2}+1\right)=\prod_{l=t}^{\binom{t}{2}-1}\left(r-l\right),$

therefore, the inequality (8) holds if and only if the following inequality
holds:

$\left((r-1)(r-3)-\frac{r\cdot
t!}{8(t-4)!}\right)r^{t(t-1)/2}-(r-1)(r-3)\left(\prod_{l=t}^{\binom{t}{2}-1}\left(r-l\right)\right)(r^{t}-r)>0.$
(9)

Consider the leading coefficient of the polynomial on the left hand side of
(9). Notice that the term with the largest power of $r$ that appears in (9) is
$r^{2+t(t-1)/2}$, however its coefficient is $0$. Therefore, we will focus on
the coefficient of the term $r^{1+t(t-1)/2}$, which is

$\left(-4-\frac{t!}{8(t-4)!}\right)-\left(-4-\left(\sum_{l=t}^{\binom{t}{2}-1}l\right)\right)=t(t-1)(t-3)/2.$
(10)

Since $t\geq 4$, the leading coefficient calculated in (10) is strictly
positive. This implies that there exists a large enough integer $r_{t}$ that
satisfies the inequality (9), in fact, the inequality (9) will hold for all
$r\geq r_{t}$. Therefore, $K_{t}$ is $r$-rainbow uncommon for all $r\geq
r_{t}$. ∎

Theorem 3 follows directly from the theorem above. Theorem 2 follows from
Proposition 1 and by substituting $t=4,r=6$ and $t=5,r=10$, into (9) and
seeing that both cases satisfy the inequality.

##  3 Conclusion

Sun very recently showed that any graph containing a cycle is $r$-rainbow-
uncommon for all $r\geq e$ [5]. While that implies our result, we explicitly
construct the colorings for complete graphs that show rainbow-uncommonness.

Future work is to investigate upper bounds, particularly to show which graphs
are $r$-rainbow-common. Thus far, the only class of graphs known to be
rainbow-common are disjoint unions of stars [4]. It is also likely that
$P_{4}$, a path on four vertices, is $3$-rainbow-common. We believe the
converse of Sun’s result is also true, formulated in the following conjecture.

######  Conjecture 1.

$H$ is rainbow-common if and only if $H$ is a forest.

By rainbow-common, we mean $r$-rainbow-common for all $r\geq e$.

## Acknowledgements

This work was done as part of the 2021 Polymath Jr program, partially
supported by NSF award DMS–2113535. We are grateful to the organizers of the
Polymath Jr. program for creating the environment to carry out this research
project. We also thank our colleagues Pablo Blanco and Sarvagya Jain who
contributed to helpful conversations and questions that were valuable for the
present paper. The work of the second author was partially supported by the
Fairfield Summer Research award.

## References

  * [1] David Conlon, Jacob Fox, and Benny Sudakov, _Recent developments in graph Ramsey theory_ , arXiv e-prints (2015), arXiv:1501.02474.
  * [2] Paul Erdős and András Hajnal, _On ramsey like theorems, problems and results_ , Combinatorica (1972).
  * [3] A. W. Goodman, _On sets of acquaintances and strangers at any party_ , The American Mathematical Monthly 66 (1959), no. 9, 778–783.
  * [4] Jessica De Silva, Xiang Si, Michael Tait, Yunus Tunçbilek, Ruifan Yang, and Michael Young, _Anti-ramsey multiplicities_ , Australas. J Comb. 73 (2019), 357–371.
  * [5] Yihang Sun, _Rainbow common graphs must be forests_ , (2023).