# Common best proximity point theorems under proximal $F$-weak dominance in
complete metric spaces

Aman Deep  Department of Mathematics, University of Delhi, Delhi-110007
<EMAIL_ADDRESS>and  Rakesh Batra<EMAIL_ADDRESS>Department of Mathematics, Hansraj College, University of Delhi, Delhi-110007

###### Abstract.

Suppose that $S_{1}$ and $S_{2}$ are nonempty subsets of a complete metric
space $(\mathcal{M},d)$ and $\phi,\psi:S_{1}\to S_{2}$ are mappings. The aim
of this work is to investigate some conditions on $\phi$ and $\psi$ such that
the two functions, one that assigns to each $x\in S_{1}$ exactly $d(x,\phi x)$
and the other that assigns to each $x\in S_{1}$ exactly $d(x,\psi x)$, attain
the global minimum value at the same point in $S_{1}$. We have introduced the
notion of proximally $F$-weakly dominated pair of mappings and proved two
theorems that guarantee the existence of such a point. Our work is an
improvement of earlier work in this direction. We have also provided examples
in which our results are applicable, but the earlier results are not
applicable.

###### Key words and phrases:

Best proximity point, $F$-dominance, $F$-weak dominance, proximal $F$-weak
dominance, fixed point

###### 2020 Mathematics Subject Classification:

54H25, 47H10, 55M20

##  1\. Introduction

After the discovery of contraction mapping theorem by Stefan Banach in 1922,
it became apparent that it was a very useful result. This prompted other
scholars towards finding similar results, and thus the vast field of metric
fixed point theory was born. For a detailed discussion on such theorems, one
may refer to [10] and the references therein.

One of the recent generalizations of contraction mapping theorem was given by
Wardowski [11], who defined the so called $F$-contraction mappings. He not
only proved that every $F$-contraction satisfies the conclusion of contraction
mapping theorem, but also produced an example of a mapping which is an
$F$-contraction but does not satisfy the hypothesis of contraction mapping
theorem. Many scholars tried to generalize the concept of an $F$-contraction
in many directions. The recent survey by Karapinar et al. [9] contains a
detailed discussion of various generalizations of $F$-contractions.

Suppose that $S_{1}$ and $S_{2}$ are nonempty subsets of a metric space
$(\mathcal{M},d)$ and $\phi:S_{1}\to S_{2}$ be a mapping. If $S_{1}\cap
S_{2}=\emptyset$ then we cannot talk about a fixed point of $\phi$. In such a
case, we search for points in $S_{1}$ such that the error $d(x,\phi x)$ is as
small as possible. The initial breakthrough in this direction was obtained by
Ky Fan in 1969, who proved the following theorem.

######  Theorem 1 ([8]).

Let $S$ be a nonempty compact convex set in a locally convex real topological
vector space $\mathcal{M}$. Let $\phi:S\to\mathcal{M}$ be a continuous
mapping. Then either $\phi$ has a fixed point in $S$ or there exists a point
$x$ in $S$ and a continuous seminorm $p$ on $\mathcal{M}$ such that

$0<p(x-\phi x)=\inf_{y\in S}p(y-\phi x).$

In the modern terminology, theorems which guarantee a conclusion similar to
that of Theorem 1 are called best approximation theorems. On the other hand,
the theorems which guarantee the existence of a point $x\in S_{1}$ such that
$d(x,\phi x)=d(S_{1},S_{2})=\inf\\{d(s_{1},s_{2}):s_{1}\in S_{1},s_{2}\in
S_{2}\\}$ are called best proximity point theorems. Some early work in this
direction was due to Anuradha and Veeramani [2], Saddiq Basha [3] and Eldred
et al. [7].

Recently, R. Batra [5] defined the so called proximally $F$-dominated pair of
mappings and proved a common best proximity point theorem for such mappings.
In doing so, he generalized the earlier work of Saddiq Basha [4]. The main
objective of our paper is to considerably weaken the notion of proximally
$F$-dominated pair of mappings and to prove common best proximity point
theorems for such mappings.

##  2\. Preliminaries

We shall use the symbols $\mathbb{N}$, $\mathbb{R}$ and $\mathbb{R_{+}}$ to
denote the set of non-negative integers, the set of real numbers and the set
of positive real numbers, respectively. Convergence of a sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}$ to $x$ as $i\to\infty$ in a metric space will
often be denoted by $x_{i}\to x$. The value of a function $\phi$ at $x$ will
be denoted by $\phi x$ instead of $\phi(x)$ when there is no possibility of
confusion. If $S_{1}$ and $S_{2}$ are sets, then the expression $S_{1}\subset
S_{2}$ implies that $S_{1}$ is a subset of $S_{2}$. If $\phi:S_{1}\to S_{2}$
and $\psi:S_{3}\to S_{4}$ are functions such that $S_{2}\subset S_{3}$, then,
$\psi\circ\phi:S_{1}\to S_{4}$ denotes the composition of functions, defined
by $\psi\circ\phi x=\psi(\phi x)$ for all $x\in S_{1}$. The composition of a
function $\phi:S\to S$ with itself $n$ times will be denoted by $\phi^{n}$,
with the understanding that $\phi^{0}$ is the identity mapping of $S$. If
$\phi:S_{1}\to\ S_{2}$ is a function, and $S\subset S_{1}$ then we denote by
$\phi_{|S}$ the mapping $\phi_{|S}:S\to\phi(S)$ defined by $\phi_{|S}x=\phi x$
for all $x\in S$.

Let $S_{1}$ and $S_{2}$ be nonempty subsets of a metric space
$(\mathcal{M},d)$. Following S. Basha [4], we shall use the notations:
$d(S_{1},S_{2})$ as the infimum of the set of all possible distances between
two points, one each from $S_{1}$ and $S_{2}$; and $S_{1}^{0}$(respectively,
$S_{2}^{0}$) as the set of all $s_{1}\in S_{1}$(respectively, $s_{2}\in
S_{2}$) such that $d(S_{1},S_{2})$ is attained by $d$ at $(s_{1},s_{2})$ for
some $s_{2}\in S_{2}$(respectively, $s_{1}\in S_{1}$).

Now, we recall a well known notion of a closed mapping.

######  Definition 1 ([1] p. 51).

Let $(\mathcal{M}_{1},d_{1})$ and $(\mathcal{M}_{2},d_{2})$ be metric spaces
and $\phi:\mathcal{M}_{1}\to\mathcal{M}_{2}$ be a mapping. We say that $\phi$
is closed if for every sequence $\\{x_{i}\\}_{i\in\mathbb{N}}$ in
$\mathcal{M}_{1}$,

$[x_{i}\to x\in\mathcal{M}_{1}\text{ and }\phi x_{i}\to
y\in\mathcal{M}_{2}]\Rightarrow\phi x=y.$

######  Remark 1 ([1] p. 51).

It is well known that every continuous mapping is closed, but the converse is
not true.

The notion of proximal commutativity will play a prominent role throughout the
paper.

######  Definition 2 ([4]).

Let $S_{1}$ and $S_{2}$ be nonempty subsets of a metric space
$(\mathcal{M},d)$. A pair of mappings $\phi,\psi:S_{1}\to S_{2}$ is said to be
proximally commuting, if

$\left[d(a,\phi x)=d(b,\psi x)=d(S_{1},S_{2})\right]\Rightarrow\phi b=\psi a$

for all $a,b,x\in S_{1}$.

It is easy to see that if $S_{1}=S_{2}$, then $\phi$ and $\psi$ are proximally
commuting if and only if they are commuting. Therefore, proximal commutativity
is a generalization of usual commutativity.

Now, we recall the definition of common best proximity point.

######  Definition 3 ([4]).

Given any pair of nonempty subsets $S_{1}$ and $S_{2}$ of a metric space
$(\mathcal{M},d)$ and a pair of mappings $\phi,\psi:S_{1}\to S_{2}$, a point
$x^{*}\in S_{1}$ is called a common best proximity point of $\phi$ and $\psi$
if $d(x^{*},\phi x^{*})=d(S_{1},S_{2})$ and $d(x^{*},\psi
x^{*})=d(S_{1},S_{2})$.

Since, the mappings $x\to d(x,\phi x)$ and $x\to d(x,\psi x)$ are generally
nonlinear and $d(S_{1},S_{2})\leq d(x,\phi x)$ for all $x\in S_{1}$, hence, a
common best proximity point problem may be regarded as a global nonlinear
optimization problem. In [11], Wardowski used a class of functions
$F:\mathbb{R}_{+}\to\mathbb{R}$ satisfying the following properties to define
the so called $F$-contractions.

  1. (F1)

$F$ is strictly increasing i.e. $x<y\Rightarrow Fx<Fy$ for all
$x,y\in\mathbb{R}_{+}$;

  2. (F2)

for every sequence $\\{\alpha_{i}\\}_{i\in\mathbb{N}}$ of positive real
numbers, $\alpha_{i}\to 0\iff F\alpha_{i}\to-\infty$;

  3. (F3)

there exists $k\in(0,1)$ such that $\lim_{\alpha\to
0^{+}}\alpha^{k}F\alpha=0.$

We denote by $\mathcal{F}$ the class of all mappings
$F:\mathbb{R}_{+}\to\mathbb{R}$ satisfying the conditions (F1)-(F3).

######  Example 1 ([11]).

Consider the mappings $F_{i}:\mathbb{R}_{+}\to\mathbb{R}(i=1,2,3,4)$ defined
by

  1. (i)

$F_{1}\alpha=\ln{\alpha}$;

  2. (ii)

$F_{2}\alpha=\ln{\alpha}+\alpha$;

  3. (iii)

$F_{3}\alpha=\frac{-1}{\sqrt{\alpha}}$;

  4. (iv)

$F_{4}\alpha=\ln(\alpha^{2}+\alpha).$

It was ensured by Wardowski that $F_{1},F_{2},F_{3},F_{4}\in\mathcal{F}$.

R. Batra [5] defined the concept of proximally $F$-dominated pair of mappings
as follows:

######  Definition 4 ([5]).

Given any $F\in\mathcal{F}$ and two nonempty subsets $S_{1}$ and $S_{2}$ of a
metric space $(\mathcal{M},d)$, a function $\psi:S_{1}\to S_{2}$ is said to
$F$-dominate another function $\phi:S_{1}\to S_{2}$ proximally, if there
exists a real number $\tau>0$ such that

$\left.\begin{array}[]{rll}d(a_{1},\phi x_{1})=d(a_{2},\phi
x_{2})&=d(S_{1},S_{2})\\\ d(b_{1},\psi x_{1})=d(b_{2},\psi
x_{2})&=d(S_{1},S_{2})\\\ &a_{1}\neq
a_{2}\end{array}\right\\}\Rightarrow\left\\{\begin{array}[]{ll}&b_{1}\neq
b_{2}\\\ &\tau+F(d(a_{1},a_{2}))\leq F(d(b_{1},b_{2}))\end{array}\right.$

for all $a_{1},a_{2},b_{1},b_{2},x_{1},x_{2}\in S_{1}$.

For all $a_{1},a_{2},b_{1},b_{2}\in S_{1}$, let us define

$M_{d}(a_{1},a_{2},b_{1},b_{2})=\max\left\\{d(b_{1},b_{2}),d(a_{2},b_{2}),d(a_{1},b_{1}),\frac{d(a_{2},b_{1})+d(a_{1},b_{2})}{2}\right\\}.$

In [12], Wardowski and Dung introduced the concept of $F$-weak contractions as
follows:

######  Definition 5 ([12]).

Let $(\mathcal{M},d)$ be a metric space and $F\in\mathcal{F}$. Let
$\phi:\mathcal{M}\to\mathcal{M}$. We say that $\phi$ is an $F$-weak
contraction if there exists $\tau>0$ such that

$\phi x_{1}\neq\phi x_{2}\Rightarrow\tau+F(d(\phi x_{1},\phi x_{2}))\leq
F(M_{d}(\phi x_{1},\phi x_{2},x_{1},x_{2}))$

for all $x_{1},x_{2}\in\mathcal{M}$.

The following important generalization of contraction mapping theorem was also
proved in [12].

######  Theorem 2 ([12]).

Let $F\in\mathcal{F}$ and $(\mathcal{M},d)$ be a complete metric space. Let
$\phi:\mathcal{M}\to\mathcal{M}$ be an $F$-weak contraction. If $\phi$ or $F$
is continuous, then $\phi$ has a unique fixed point $x^{*}$ in $\mathcal{M}$.
Moreover, for any $x_{0}\in\mathcal{M}$, we have
$\lim_{i\to\infty}\phi^{i}x_{0}=x^{*}$.

##  3\. Main Results

Taking inspiration from [12], we define the concept of proximally $F$-weakly
dominated pair of mappings as follows:

######  Definition 6.

Given any $F\in\mathcal{F}$ and two nonempty subsets $S_{1}$ and $S_{2}$ of a
metric space $(\mathcal{M},d)$, a function $\psi:S_{1}\to S_{2}$ is said to
$F$-weakly dominate another function $\phi:S_{1}\to S_{2}$ proximally, if
there exists a real number $\tau>0$ such that

$\left.\begin{array}[]{rll}d(a_{1},\phi x_{1})=d(a_{2},\phi
x_{2})&=d(S_{1},S_{2})\\\ d(b_{1},\psi x_{1})=d(b_{2},\psi
x_{2})&=d(S_{1},S_{2})\\\ &a_{1}\neq a_{2}\end{array}\right\\}\\\
\Rightarrow\left\\{\begin{array}[]{ll}&M_{d}(a_{1},a_{2},b_{1},b_{2})\neq 0\\\
&\tau+F(d(a_{1},a_{2}))\leq
F(M_{d}(a_{1},a_{2},b_{1},b_{2}))\end{array}\right.$

for all $a_{1},a_{2},b_{1},b_{2},x_{1},x_{2}\in S_{1}$.

######  Proposition 1.

For nonempty subsets $S_{1}$ and $S_{2}$ of a metric space $(\mathcal{M},d)$
and $F\in\mathcal{F}$, if $\phi:S_{1}\to S_{2}$ is $F$-dominated by another
mapping ${\psi:S_{1}\to S_{2}}$ proximally, then $\phi$ is also $F$-weakly
dominated by $\psi$ proximally.

###### Proof.

The proposition follows trivially from the fact that $F$ is strictly
increasing and that $M_{d}(a_{1},a_{2},b_{1},b_{2})\geq d(b_{1},b_{2})$ for
all ${a_{1},a_{2},b_{1},b_{2}\in S_{1}}$. ∎

Example 6 shows that the converse of Proposition 1 may not be true. Hence, for
a fixed pair of subsets $(S_{1},S_{2})$ of a metric space $(\mathcal{M},d)$,
the class of all pairs of mappings $(\phi,\psi)$ such that $\phi$ is
$F$-dominated by $\psi$ proximally for some $F\in\mathcal{F}$ is generally a
proper subclass of the class of all pairs of mappings $(\phi,\psi)$ such that
$\phi$ is $F$-weakly dominated by $\psi$ proximally for some
$F\in\mathcal{F}$.

If $S_{1}=S_{2}=\mathcal{M}$, then the concept of best proximity point reduces
to the concept of fixed point. The concept of $F$-weakly dominated pair of
mappings proximally as introduced in Definition 6 reduces to the following.

######  Definition 7.

Let $(\mathcal{M},d)$ be a metric space and $F\in\mathcal{F}$. Let $\phi$ and
$\psi$ be functions mapping $\mathcal{M}$ into $\mathcal{M}$. Then, $\psi$ is
said to $F$-weakly dominate $\phi$ if there exists $\tau>0$ such that

$\phi x_{1}\neq\phi x_{2}\Rightarrow\left\\{\begin{array}[]{ll}&M_{d}(\phi
x_{1},\phi x_{2},\psi x_{1},\psi x_{2})\neq 0\\\ &\tau+F(d(\phi x_{1},\phi
x_{2}))\leq F(M_{d}(\phi x_{1},\phi x_{2},\psi x_{1},\psi
x_{2}))\end{array}\right.$

for all $x_{1},x_{2}\in\mathcal{M}$.

A definition similar to the Definition 7 was first introduced by Zhou et al.
in [13] under the name “Ćirić-type $F_{M}$-contraction”, however, the authors
assume the continuity of $F$ in their definition. So, Definition 7 is more
general and is also an improvement of Definition 3.2 in [6], which is a
special case of Definition 4 with $S_{1}=S_{2}=\mathcal{M}.$

The following proposition gives the relationship between $F$-weak contractions
and $F$-weakly dominated pair of mappings.

######  Proposition 2.

Let $(\mathcal{M},d)$ be a metric space and $F\in\mathcal{F}$. Let
$\phi:\mathcal{M}\to\mathcal{M}$ be a mapping and
$I:\mathcal{M}\to\mathcal{M}$ be the identity mapping defined by $Ix=x$ for
all $x\in\mathcal{M}$. Then, $\phi$ is $F$-weakly dominated by $I$ if and only
if $\phi$ is an $F$-weak contraction.

Therefore, the concept of $F$-weakly dominated pair of mappings can be
considered a broad generalization of the concept of $F$-weak contractions.

Before proving the best proximity point theorems, we prove the following
technical lemmas to avoid the repetition of arguments later.

######  Lemma 1.

Let $F\in\mathcal{F}$ and $(\mathcal{M},d)$ be a complete metric space.
Consider any two nonempty subsets $S_{1}$ and $S_{2}$ of $\mathcal{M}$ with
$S_{1}^{0}$ as nonempty and closed. Let ${\phi,\psi:S_{1}\to S_{2}}$ be a pair
of proximally commuting mappings which satisfy the following conditions:

  1. (a)

$\psi$ is such that it $F$-weakly dominates $\phi$ proximally;

  2. (b)

$\phi(S_{1}^{0})\subset S_{2}^{0}$;

  3. (c)

$\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$

then, there exists a sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ in $S_{1}^{0}$
which satisfies the following properties:

  1. (1)

$\phi a_{i}=\psi a_{i+1}$ for all $i\in\mathbb{N}$;

  2. (2)

if $a_{i}\neq a_{i+1}$ for all $i\in\mathbb{N}$, then there exists $a\in
S_{1}^{0}$ such that $a_{i}\to a$.

###### Proof.

Choose $x_{0}\in S_{1}^{0}$ arbitrarily. Since
$\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$, we may define a sequence
$\\{x_{i}\\}_{i\in\mathbb{N}}$ of points of $S_{1}^{0}$ inductively satisfying

(1) $\phi x_{i}=\psi x_{i+1}$

for all $i\in\mathbb{N}$. Since $\phi(S_{1}^{0})\subset S_{2}^{0}$, we can
find a sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ in $S_{1}^{0}$ such that

(2) $d(\phi x_{i},a_{i})=d(S_{1},S_{2})$

for all $i\in\mathbb{N}$. The equations (1) and (2) give the following system
of equations:

(3) $\left.\begin{array}[]{cc}&d(\phi x_{i+1},a_{i+1})=d(S_{1},S_{2})\\\
&d(\psi x_{i+1},a_{i})=d(S_{1},S_{2})\end{array}\right\\}$

for all $i\in\mathbb{N}$. The system (3) along with the proximal commutativity
of $\phi$ and $\psi$ gives

$\phi a_{i}=\psi a_{i+1}$

for all $i\in\mathbb{N}$.

Assume that $a_{i}\neq a_{i+1}$ for all $i\in\mathbb{N}$. We claim that
$\\{a_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence. Suppose that $i\geq 1$.
The equations (1) and (2) give

(4) $\left.\begin{array}[]{cc}&d(\phi x_{i+1},a_{i+1})=d(\phi
x_{i},a_{i})=d(S_{1},S_{2})\\\ &d(\psi x_{i+1},a_{i})=d(\psi
x_{i},a_{i-1})=d(S_{1},S_{2})\\\ &\quad\quad a_{i}\neq
a_{i+1}.\end{array}\right\\}$

The system (4) along with the hypothesis (a) imply that there exists $\tau>0$
such that

$\begin{split}&F(d(a_{i},a_{i+1}))\leq
F(d(M_{d}(a_{i+1},a_{i},a_{i},a_{i-1})))-\tau\\\
&=F\biggl{(}\max\biggl{\\{}d(a_{i},a_{i-1}),d(a_{i},a_{i-1}),d(a_{i+1},a_{i}),\\\
&\qquad\qquad\qquad\quad\quad\qquad\qquad\quad\left.\left.\frac{d(a_{i},a_{i})+d(a_{i+1},a_{i-1})}{2}\right\\}\right)-\tau\\\
&=F\left(\max\left\\{d(a_{i},a_{i-1}),d(a_{i-1},a_{i}),d(a_{i},a_{i+1}),\frac{d(a_{i+1},a_{i-1})}{2}\right\\}\right)-\tau\\\
&=F\left(\max\left\\{d(a_{i},a_{i-1}),d(a_{i},a_{i+1}),\frac{d(a_{i+1},a_{i-1})}{2}\right\\}\right)-\tau\\\
&\leq
F\left(\max\left\\{d(a_{i},a_{i-1}),d(a_{i},a_{i+1}),\frac{d(a_{i},a_{i+1})+d(a_{i-1},a_{i})}{2}\right\\}\right)-\tau\\\
&=F\left(\max\left\\{d(a_{i},a_{i+1}),d(a_{i},a_{i-1})\right\\}\right)-\tau.\end{split}$

In above calculations we took the advantage of the monotonicity of $F$ and the
triangle inequality. Hence,

(5) $F(d(a_{i},a_{i+1}))\leq
F(\max\\{d(a_{i},a_{i+1}),d(a_{i},a_{i-1})\\})-\tau.$

If there exists $i\in\mathbb{N},i\geq 1$ such that

$\max\\{d(a_{i},a_{i+1}),d(a_{i},a_{i-1})\\}=d(a_{i},a_{i+1}),$

then, by inequality (5) we obtain

$F(d(a_{i},a_{i+1}))\leq F(d(a_{i},a_{i+1}))-\tau.$

Which is not true. Hence, we must have

(6) $\max\left\\{d(a_{i},a_{i-1}),d(a_{i},a_{i+1})\right\\}=d(a_{i},a_{i-1}).$

The inequality (5) and the equation (6) give

(7) $\begin{split}F(d(a_{i},a_{i+1}))&\leq F(d(a_{i-1},a_{i}))-\tau\\\ &\leq
F(d(a_{i-2},a_{i-1}))-2\tau\\\ &\leq F(d(a_{i-3},a_{i-2}))-3\tau\\\
&\ldots\ldots\ldots\ldots\ldots\ldots\\\ &\leq
F(d(a_{0},a_{1}))-i\tau\end{split}$

for all $i\in\mathbb{N},i\geq 1$. The inequality (7) implies that

$\lim_{i\to\infty}F(d(a_{i},a_{i+1}))=-\infty.$

Using (F2) it follows that

$\lim_{i\to\infty}d(a_{i},a_{i+1})=0.$

By (F3) there exists $k\in(0,1)$ such that

(8) $\lim_{i\to\infty}(d(a_{i},a_{i+1}))^{k}F(d(a_{i},a_{i+1}))=0.$

Note that

(9)
$\begin{split}&d(a_{i},a_{i+1})^{k}F(d(a_{0},a_{1}))-d(a_{i},a_{i+1})^{k}F(d(a_{i},a_{i+1}))\\\
&\geq
d(a_{i},a_{i+1})^{k}F(d(a_{0},a_{1}))-d(a_{i},a_{i+1})^{k}(F(d(a_{0},a_{1}))-i\tau)\\\
&=id(a_{i},a_{i+1})^{k}\tau\geq 0.\end{split}$

Using (8) and (9) we obtain

$\lim_{i\to\infty}i(d(a_{i},a_{i+1}))^{k}=0.$

Therefore, we can find $i_{0}\in\mathbb{N}$ such that

$\begin{split}&i(d(a_{i},a_{i+1}))^{k}\leq 1\text{ for all }i\geq i_{0}\\\
&\Rightarrow d(a_{i},a_{i+1})\leq\frac{1}{i^{1/k}}\text{ for all }i\geq
i_{0}.\end{split}$

Let $j,i\in\mathbb{N}$ be such that $j>i\geq i_{0}$. Therefore,

$\begin{split}d(a_{j},a_{i})&\leq d(a_{j},a_{j-1})+\ldots+d(a_{i+1},a_{i})\\\
&<\sum_{l=i}^{\infty}d(a_{l},a_{l+1})\leq\sum_{l=i}^{\infty}\frac{1}{l^{1/k}}.\end{split}$

Since the series $\sum_{l=1}^{\infty}\frac{1}{l^{1/k}}$ converges, it follows
that $\\{a_{i}\\}_{i\in\mathbb{N}}$ is a Cauchy sequence. But $S_{1}^{0}$,
being a closed subset of a complete metric space, is itself a complete metric
space. Hence, there exists $a\in S_{1}^{0}$ such that

$\lim_{i\to\infty}a_{i}=a.$

∎

The Example 4 shows that hypothesis (a) of Lemma 1 cannot be dropped.

######  Example 2.

Let $(\mathcal{M},d)$ be a metric space. Consider the cartesian product
$\mathcal{M}^{\prime}=\mathbb{R}\times\mathcal{M}$ under the metric

$d^{\prime}((u,v),(t,w))=\sqrt{\left|u-t\right|^{2}+d(v,w)^{2}}$

for all $u,t\in\mathbb{R}$ and $v,w\in\mathcal{M}$. Define,

$S_{1}=\\{(-1,x):x\in\mathcal{M}\\}$ $S_{2}=\\{(1,x):x\in\mathcal{M}\\}.$

Let $\phi^{\prime},\psi^{\prime}:\mathcal{M}\to\mathcal{M}$ be mappings. We
define $\phi,\psi:S_{1}\to S_{2}$ as follows:

$\phi(-1,x)=(1,\phi^{\prime}x)$ $\psi(-1,x)=(1,\psi^{\prime}x).$

It is easy to check that $\phi$ and $\psi$ commute proximally if and only if
$\phi^{\prime}$ and $\psi^{\prime}$ commute.

######  Example 3.

Let $\mathcal{M}=\mathbb{C}$ under the metric $d(z,w)=|z-w|$. Let $0<a<b$ be
real numbers. Define

$S_{1}=\\{z\in\mathbb{C}:|z|=a\\}$

and

$S_{2}=\\{z\in\mathbb{C}:|z|=b\\}.$

For any point $z$ on $S_{1}$, draw a diameter of $S_{2}$ passing through $z$
as shown in Figure 1. The diameter cuts the circle $S_{2}$ at two points.
Denote the point nearer to $z$ by $\phi z$ and the other point by $\psi z$. It
is easy to see that the mappings $\phi,\psi:S_{1}\to S_{2}$ commute
proximally.

$S_{1}$$S_{2}$$0$$z$$\phi z$$\psi z$

Figure 1. Geometrical description of the mappings $\phi$ and $\psi$

######  Example 4.

Let $\mathcal{M},d,S_{1},S_{2},\phi,\psi$ be as in Example 3. Clearly, in this
case $S_{1}=S_{1}^{0}$ and $S_{2}=S_{2}^{0}$. It was discussed in Example 3
that $\phi$ and $\psi$ commute proximally. It is easy to check that the
hypotheses (b) and (c) of Lemma 1 are also satisfied. To see that the
hypothesis (a) is not satisfied, let $z_{1}$ and $z_{2}$ be points in $S_{1}$
that lie on the opposite ends of a diameter of $S_{1}$. Take
$a_{1}=b_{2}=z_{1}$ and $a_{2}=b_{1}=z_{2}$. We have

$\left.\begin{array}[]{cc}&d(a_{1},\phi z_{1})=d(a_{2},\phi
z_{2})=d(S_{1},S_{2})\\\ &d(b_{1},\psi z_{1})=d(b_{2},\psi
z_{2})=d(S_{1},S_{2})\\\ &\quad\quad a_{1}\neq a_{2}.\end{array}\right\\}$

Let if possible $\phi$ be $F$-weakly dominated by $\psi$ proximally for some
$F\in\mathcal{F}$. Then, we can find some $\tau>0$ such that

$\begin{split}\tau+F(d(a_{1},a_{2}))&\leq F(M_{d}(a_{1},a_{2},b_{1},b_{2}))\\\
&=F(d(a_{1},a_{2})).\end{split}$

Which is a contradiction.

Let, if possible, the conclusion of Lemma 1 be satisfied. Then, there exists a
sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ in $S_{1}$ such that

$\phi a_{i}=\psi a_{i+1}$

for all $i\in\mathbb{N}$. Since $\phi$ and $\psi$ cannot satisfy the condition
$\phi a=\psi a$ for any $a\in S_{1}$, hence, $a_{i}\neq a_{i+1}$ for all
$i\in\mathbb{N}$. Thus, the sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ must
converge to some $a\in S_{1}$. But, since $\phi$ and $\psi$ are continuous,
this implies that $\phi a=\psi a$, which is again a contradiction.

######  Lemma 2.

Let $S_{1}$ and $S_{2}$ be nonempty subsets of a metric space
$(\mathcal{M},d)$. Let $F\in\mathcal{F}$ and the mappings $\phi,\psi:S_{1}\to
S_{2}$ be a pair of proximally commuting mappings which satisfy the following
conditions:

  1. (a)

$\psi$ is such that it $F$-weakly dominates $\phi$ proximally;

  2. (b)

$\phi(S_{1}^{0})\subset S_{2}^{0}$;

  3. (c)

there exists $x\in S_{1}^{0}$ such that $\phi x=\psi x$

then, $\phi$ and $\psi$ have a unique common best proximity point.

###### Proof.

By hypothesis (b), hypothesis (c) and the definition of $S_{2}^{0}$, there
exists some $a\in S_{1}^{0}$ such that

(10) $\left.\begin{array}[]{cc}&d(a,\phi x)=d(S_{1},S_{2})\\\ &d(a,\psi
x)=d(S_{1},S_{2}).\end{array}\right\\}$

The proximal commutativity of $\phi$ and $\psi$ along with the system (10)
implies

$\phi a=\psi a.$

Again, as $\phi(S_{1}^{0})\subset S_{2}^{0}$, we can find $b\in S_{1}^{0}$
such that

(11) $\left.\begin{array}[]{cc}&d(b,\phi a)=d(S_{1},S_{2})\\\ &d(b,\psi
a)=d(S_{1},S_{2}).\end{array}\right\\}$

We claim that $b=a$. Suppose not, hence from (10) and (11) we obtain the
following system

(12) $\left.\begin{array}[]{cc}&d(a,\phi x)=d(b,\phi a)=d(S_{1},S_{2})\\\
&d(a,\psi x)=d(b,\psi a)=d(S_{1},S_{2})\\\ &\quad\quad a\neq
b.\end{array}\right\\}$

System (12) along with hypothesis (a) implies that there exists some $\tau>0$
such that

$\tau+F(d(a,b))\leq F(M_{d}(a,b,a,b))=F(d(a,b))$

which is not true. So $a=b$. Hence, equations (11) give

$d(a,\phi a)=d(a,\psi a)=d(S_{1},S_{2}).$

Thus, $a$ is a common best proximity point of $\phi$ and $\psi$.

For uniqueness, let $a_{i}$(i=1,2) be common best proximity points of $\phi$
and $\psi$ such that $a_{1}\neq a_{2}$. Thus, we obtain the following system
of equations:

(13) $\left.\begin{array}[]{cc}&d(a_{1},\phi a_{1})=d(a_{2},\phi
a_{2})=d(S_{1},S_{2})\\\ &d(a_{1},\psi a_{1})=d(a_{2},\psi
a_{2})=d(S_{1},S_{2})\\\ &\quad\quad a_{1}\neq a_{2}.\end{array}\right\\}$

System (13) along with hypothesis (a) implies that

$\tau+F(d(a_{1},a_{2}))\leq
F(M_{d}(a_{1},a_{2},a_{1},a_{2}))=F(d(a_{1},a_{2}))$

which is not true. Hence, we must have $a_{1}=a_{2}$. ∎

For the mappings $\phi,\psi:S_{1}\to S_{2}$, a point $x\in S_{1}$ which
satisfies $\phi x=\psi x$ is called a coincidence point of $\phi$ and $\psi$.
There is a vast amount of literature available on coincidence point theorems.
If we add the hypotheses of those coincidence point theorems in Lemma 2, we
obtain a corresponding common best proximity point theorem. However, the
hypotheses of Lemma 2 are still quite strong, so it may happen that no
interesting examples of the resulting theorems exist. One may think that we
can drop some hypothesis from it to obtain a more general result. The
following example shows that at least hypothesis (a) of Lemma 2 cannot be
dropped.

######  Example 5.

Consider $\mathbb{R}$ under the usual Euclidean metric $d$. Define
$\phi,\psi:\mathbb{R}\to\mathbb{R}$ as

$\phi x=\begin{cases}-\frac{1}{x}&x\neq 0\\\ \quad 1&x=0\end{cases}$ $\psi
x=\begin{cases}-\frac{1}{x^{3}}&x\neq 0\\\ \quad 1&x=0\end{cases}$

for all $x\in\mathbb{R}$. With $S_{1}=S_{2}=\mathbb{R}$, it is easy to see
that $\phi$ and $\psi$ commute proximally, and they also satisfy the
hypotheses (b) and (c) of Lemma 2. Note that $\phi$ and $\psi$ does not have
any common best proximity point (i.e. no common fixed point in this case).

We claim that the hypothesis (a) is not satisfied. Suppose that the condition
is satisfied. Note that $\phi(0)\neq\phi(1)$. If $\phi$ is $F$-weakly
dominated by $\psi$ for some $F\in\mathcal{F}$, then by Definition 7 there
exists $\tau>0$ such that

$\begin{split}&\tau+F(|\phi(0)-\phi(1)|)\leq
F(M_{d}(\phi(0),\phi(1),\psi(0),\psi(1)))\\\ &\Rightarrow\tau+F(2)\leq
F(2)\end{split}$

which is a contradiction.

It may be interesting to look for other conditions on $\phi$ and $\psi$ which
guarantee that the existence of a coincidence point implies the existence of a
common best proximity point.

By observing the statements of Lemma 1 and Lemma 2, one may notice that if
$\phi,\psi:S_{1}\to S_{2}$ satisfy all the hypotheses of Lemma 1 along with
some additional hypotheses on $\phi$ and $\psi$ which guarantee that the point
$a$ given by Lemma 1 satisfies $\phi a=\psi a$, then by Lemma 2, $\phi$ and
$\psi$ will have a unique common best proximity point. One such hypothesis is
to assume that $\phi{{}_{|S_{1}^{0}}}$ is continuous and
$\psi{{}_{|S_{1}^{0}}}$ is closed.

######  Theorem 3.

Let $F\in\mathcal{F}$ and $(\mathcal{M},d)$ be a complete metric space.
Consider any two nonempty subsets $S_{1}$ and $S_{2}$ of $\mathcal{M}$ with
$S_{1}^{0}$ as nonempty and closed. Let $\phi,\psi:S_{1}\to S_{2}$ be a pair
of proximally commuting mappings which satisfy the following conditions:

  1. (a)

$\psi$ is such that it $F$-weakly dominates $\phi$ proximally;

  2. (b)

$\phi{{}_{|S_{1}^{0}}}$ is continuous and $\psi{{}_{|S_{1}^{0}}}$ is closed;

  3. (c)

$\phi(S_{1}^{0})\subset S_{2}^{0}$;

  4. (d)

$\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$

then, $\phi$ and $\psi$ have a unique common best proximity point.

###### Proof.

By Lemma 1, we can find a sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ in
$S_{1}^{0}$ which satisfies

(14) $\phi a_{i}=\psi a_{i+1}$

for all $i\in\mathbb{N}$. If

$a_{i}=a_{i+1}=a$

for some $i\in\mathbb{N}$, then from (14) we get

$\phi a=\psi a.$

Assume now that $a_{i}\neq a_{i+1}$ for all $i\in\mathbb{N}$. Then, by Lemma
1, there exists $a\in S_{1}^{0}$ such that the sequence
$\\{a_{i}\\}_{i\in\mathbb{N}}$ converges to $a$.

Since $\phi{{}_{|S_{1}^{0}}}$ is continuous, we obtain

$\lim_{i\to\infty}\phi a_{i}=\lim_{i\to\infty}\psi a_{i+1}=\phi
a\in\phi(S_{1}^{0})\subset\psi(S_{1}^{0}).$

Since $\psi{{}_{|S_{1}^{0}}}$ is closed we must have

$\psi a=\phi a.$

Therefore, by Lemma 2, $\phi$ and $\psi$ have a unique common best proximity
point. ∎

Another condition would be to assume that $\psi{{}_{|S_{1}^{0}}}$ is injective
(i.e. $\psi x_{1}=\psi x_{2}\Rightarrow x_{1}=x_{2}$ for all $x_{1},x_{2}\in
S_{1}^{0}$) and the mapping $~{}{\xi:S_{1}^{0}\to S_{1}^{0}}$ given by
$\xi=(\psi{{}_{|S_{1}^{0}}})^{-1}\circ\phi{{}_{|S_{1}^{0}}}$ is closed.

######  Theorem 4.

Let $F\in\mathcal{F}$ and $(\mathcal{M},d)$ be a complete metric space.
Consider any two nonempty subsets $S_{1}$ and $S_{2}$ of $\mathcal{M}$ with
$S_{1}^{0}$ as nonempty and closed. Let $\phi,\psi:S_{1}\to S_{2}$ be a pair
of proximally commuting mappings which satisfy the following conditions:

  1. (a)

$\psi$ is such that it $F$-weakly dominates $\phi$ proximally;

  2. (b)

$\psi{{}_{|S_{1}^{0}}}$ is injective;

  3. (c)

the mapping $\xi:S_{1}^{0}\to S_{1}^{0}$ given by
$\xi=(\psi{{}_{|S_{1}^{0}}})^{-1}\circ\phi{{}_{|S_{1}^{0}}}$ is closed;

  4. (d)

$\phi(S_{1}^{0})\subset S_{2}^{0}$;

  5. (e)

$\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$

then, $\phi$ and $\psi$ have a unique common best proximity point.

###### Proof.

We claim that there exists $a\in S_{1}^{0}$ such that $\phi a=\psi a$.
Clearly, it is sufficient to show that the mapping $\xi:S_{1}^{0}\to
S_{1}^{0}$ has a fixed point in $S_{1}^{0}$.

By Lemma 1, there exists a sequence $\\{a_{i}\\}_{i\in\mathbb{N}}$ in
$S_{1}^{0}$ such that

(15) $\begin{split}&\phi a_{i}=\psi a_{i+1}\\\ &\Rightarrow\xi
a_{i}=a_{i+1}\end{split}$

for all $i\in\mathbb{N}$. If $a_{i}=a_{i+1}=a$ for some $i\in\mathbb{N}$,
then, by equation (15) $\xi a=a$. Suppose that $a_{i}\neq a_{i+1}$ for all
$i\in\mathbb{N}$. By Lemma 1, there exists $a\in S_{1}^{0}$ such that

$\lim_{i\to\infty}a_{i}=a.$

Since $\xi$ is closed, from equation (15) we obtain $\xi a=a$. ∎

If $S_{1}=S_{2}=\mathcal{M}$ in Theorem 3, we obtain the following common
fixed point theorem. Similar theorems were studied by Zhou et al. in [13]. It
is also an improvement of Theorem 3.9 in [6].

######  Theorem 5.

Consider an $F\in\mathcal{F}$ and a complete metric space $(\mathcal{M},d)$.
Let $\phi,\psi:\mathcal{M}\to\mathcal{M}$ be a pair of commuting mappings
which satisfy the following conditions:

  1. (a)

$\psi$ is such that it $F$-weakly dominates $\phi$;

  2. (b)

$\phi$ is continuous and $\psi$ is closed;

  3. (c)

$\phi(\mathcal{M})\subset\psi(\mathcal{M})$

then, $\phi$ and $\psi$ have a unique common fixed point.

The following result due to R. Batra can be obtained from Theorem 3:

######  Corollary 1 ([5]).

Let $F\in\mathcal{F}$ and $(\mathcal{M},d)$ be a complete metric space.
Consider any two nonempty subsets $S_{1}$ and $S_{2}$ of $\mathcal{M}$ with
$S_{1}^{0}$ as non-empty and closed. Let $\phi,\psi:S_{1}\to S_{2}$ be a pair
of proximally commuting mappings which satisfy the following conditions:

  1. (a)

$\psi$ is such that it $F$-dominates $\phi$ proximally;

  2. (b)

$\psi$ and $\phi$ both are continuous;

  3. (c)

$\phi(S_{1}^{0})\subset S_{2}^{0}$;

  4. (d)

$\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$

then, $\phi$ and $\psi$ have a unique common best proximity point.

###### Proof.

If $\phi$ and $\psi$ are continuous, then $\phi{{}_{|S_{1}^{0}}}$ and
$\psi{{}_{|S_{1}^{0}}}$ are both continuous. By Remark 1,
$\psi{{}_{|S_{1}^{0}}}$ is closed. By Proposition 1, $\phi$ is $F$-weakly
dominated by $\psi$ proximally. Hence, by Theorem 3, $\phi$ and $\psi$ have a
unique common best proximity point. ∎

The following example shows that Theorem 3 is a proper generalization of
Corollary 1.

######  Example 6.

Take $\mathcal{M}=\\{x\in\mathbb{R}:x\leq-1\text{ or }x\geq 1\\}$ in Example 2
with the Euclidean metric $d(x,y)=|x-y|$, so that
$(\mathcal{M}^{\prime},d^{\prime})$ becomes a closed subset of the complete
metric space $\mathbb{R}^{2}$ under the usual Euclidean metric of
$\mathbb{R}^{2}$. Take $\phi^{\prime},\psi^{\prime}:\mathcal{M}\to\mathcal{M}$
as follows:

$\phi^{\prime}x=\begin{cases}3&x\leq-1\\\ 5&x\geq 1\end{cases}$

and

$\psi^{\prime}x=x.$

Let $S_{1},S_{2},\phi$ and $\psi$ be as defined in Example 2. Note that
$d(S_{1},S_{2})=2$. Clearly $S_{1}=S_{1}^{0}$ and $S_{2}=S_{2}^{0}$. Since
$\phi^{\prime}$ and $\psi^{\prime}$ commute, hence, as we discussed in Example
2, $\phi$ and $\psi$ commute proximally. Also, $~{}{\phi(S_{1}^{0})\subset
S_{2}^{0}}$ and $\phi(S_{1}^{0})\subset\psi(S_{1}^{0})$. Clearly,
$\phi{{}_{|S_{1}^{0}}}$ and $\psi{{}_{|S_{1}^{0}}}$ are both continuous (and
thus, $\psi{{}_{|S_{1}^{0}}}$ is closed).

First, we show that $\phi$ is $F_{1}$-weakly dominated by $\psi$ proximally,
where $F_{1}$ is the natural logarithm function. To see this, suppose that
$(-1,a_{i}),(-1,b_{i}),$ $(-1,x_{i})\in S_{1}(i=1,2)$ be the points which
satisfy the equations:

(16)
$\left.\begin{array}[]{cc}&d^{\prime}((-1,a_{1}),\phi(-1,x_{1}))=d^{\prime}((-1,a_{2}),\phi(-1,x_{2}))=2\\\
&d^{\prime}((-1,b_{1}),\psi(-1,x_{1}))=d^{\prime}((-1,b_{2}),\psi(-1,x_{2}))=2\\\
&\quad\quad(-1,a_{1})\neq(-1,a_{2}).\end{array}\right\\}$

Equations (16) imply that

(17)
$\left.\begin{array}[]{cc}&a_{1}=\phi^{\prime}b_{1},a_{2}=\phi^{\prime}b_{2}\\\
&a_{1}\neq a_{2}.\end{array}\right\\}$

Clearly, equations(17) imply that either

$b_{1}\leq-1,b_{2}\geq 1\text{ or }b_{2}\leq-1,b_{1}\geq 1.$

First, we assume that $b_{1}\leq-1$ and $b_{2}\geq 1$. It follows that

(18)
$\begin{split}&\frac{d^{\prime}((-1,a_{1}),(-1,a_{2}))}{M_{d^{\prime}}((-1,a_{1}),(-1,a_{2}),(-1,b_{1}),(-1,b_{2}))}\\\
&=\frac{d(a_{1},a_{2})}{M_{d}(a_{1},a_{2},b_{1},b_{2})}\\\
&\leq\frac{d(a_{1},a_{2})}{d(a_{1},b_{1})}=\frac{d(\phi^{\prime}b_{1},\phi^{\prime}b_{2})}{d(\phi^{\prime}b_{1},b_{1})}\\\
&=\frac{|3-5|}{|3-b_{1}|}=\frac{2}{|3-b_{1}|}\leq\frac{2}{4}=\frac{1}{2}.\end{split}$

If $b_{2}\leq-1$ and $b_{1}\geq 1$, we can prove that the inequality (18)
still holds by using $M_{d}(a_{1},a_{2},b_{1},b_{2})\geq d(a_{2},b_{2})$.
Inequality (18) gives,

$\ln(2)+\ln(d^{\prime}((-1,a_{1}),(-1,a_{2})))\\\
\leq\ln(M_{d^{\prime}}((-1,a_{1}),(-1,a_{2}),(-1,b_{1}),(-1,b_{2}))).$

Hence, $\phi$ is $F_{1}$-weakly dominated by $\psi$ proximally with
$\tau=\ln(2)$. Now, we claim that $\phi$ is not $F$-dominated by $\psi$
proximally for any $F\in\mathcal{F}$. Note that
$a_{1}=3,a_{2}=5,b_{1}=x_{1}=-1,b_{2}=x_{2}=1$ satisfy the system of equations
(16). If $\phi$ is $F$ dominated by $\psi$ proximally for some
$F\in\mathcal{F}$, then there must exist some $\tau>0$ such that

$\begin{split}&\tau+F(d^{\prime}((-1,3),(-1,5)))\leq
F(d^{\prime}((-1,-1),(-1,1)))\\\ &\Rightarrow\tau+F(2)\leq F(2)\\\
\end{split}$

which is a contradiction. Of course, the unique common best proximity point of
$\phi$ and $\psi$, as guaranteed by Theorem 3 is $(-1,5)$.

##  4\. Acknowledgement

As a Ph.D. student, the first author is grateful to the University of Delhi
and the Government of India for awarding him with a Non N.E.T. fellowship.

##  5\. Conflict of interest

Authors declare that there is no conflict of interest regarding the
publication of this article.

##  6\. Authors’ contribution

Both authors contributed equally in writing this article. Both authors read
and approved the final manuscript.

## References

  * [1] Charalambos Aliprantis and Kim Border  “Infinite Dimensional Analysis: A Hitchhiker’s Guide”  Springer-Verlag, 2006, pp. 703  DOI: 10.1007/3-540-29587-9
  * [2] J Anuradha and P Veeramani  “Proximal pointwise contraction”  In _Topology and its Applications_ 156.18 Elsevier, 2009, pp. 2942–2948
  * [3] S Sadiq Basha  “Best proximity points: global optimal approximate solutions”  In _Journal of Global Optimization_ 49.1 Springer, 2011, pp. 15–21
  * [4] S Sadiq Basha  “Common best proximity points: global minimal solutions”  In _Top_ 21.1 Springer, 2013, pp. 182–188
  * [5] Rakesh Batra  “Common best proximity points for proximally F-dominated mappings”  In _Fasciculi Mathematici_ 63, 2020  DOI: 10.21008/j.0044-4413.2020.0002
  * [6] Rakesh Batra  “Common fixed points for F-dominated mappings”  In _Global Journal of Pure and Applied Mathematics_ 13.3, 2017, pp. 885–892
  * [7] A Eldred, W Kirk and P Veeramani  “Proximal normal structure and relatively nonexpansive mappings”  In _Studia Mathematica_ 3.171, 2005, pp. 283–293
  * [8] Ky Fan  “Extensions of two fixed point theorems of FE Browder”  In _Mathematische zeitschrift_ 112.3 Springer, 1969, pp. 234–240
  * [9] Erdal Karapınar, Andreea Fulga and Ravi P Agarwal  “A survey:F-contractions with related fixed point results”  In _Journal of Fixed Point Theory and Applications_ 22.3 Springer, 2020, pp. 1–58
  * [10] W. Kirk and Brailey Sims  “Handbook of Metric Fixed Point Theory”  Springer, Dordrecht, 2001, pp. XIV704  DOI: 10.1007/978-94-017-1748-9
  * [11] Dariusz Wardowski  “Fixed points of a new type of contractive mappings in complete metric spaces”  In _Fixed point theory and applications_ 2012.1 SpringerOpen, 2012, pp. 1–6
  * [12] Dariusz Wardowski and Nguyen Dung  “Fixed Points Of F-Weak Contractions On Complete Metric Spaces”  In _Demonstratio Mathematica [electronic only]_ 47, 2014  DOI: 10.2478/dema-2014-0012
  * [13] Mi Zhou, Xiao-Lan Liu and Adrian Secelean  “On coincidence and common fixed point theorems of eight self-maps satisfying an FM-contraction condition”  In _Nonlinear Analysis: Modelling and Control_ 24.6, 2019, pp. 1001–1018