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\renewcommand\datereceived{August 22, 2013}
\renewcommand\dateaccepted{September 24, 2013}

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\markboth{R.\,K.\,Bisht and V.\,Rako\v cevi\' c}{Some notes on PD-operator pairs}

\title{Some notes on  PD-operator pairs\thanks{The second author is supported by grant No. 174025 of the
Ministry of Science, Technology and Development, Republic of Serbia.}}

\author[R.\,K.\,Bisht and V.\,Rako\v cevi\' c]{Ravindra K.\,Bisht\affil{1}\comma\corrauth and Vladimir Rako\v cevi\' c\affil{2}}

\address{\affilnum{1}\ Department of Mathematics, Applied Sciences and Humanities, Uttarakhand Technical University,
Bipin Tripahti Kumaon Institute of Technology, Dwarahat-262\,553 Almora, India\\
\affilnum{2}\ Faculty of Sciences and Mathematics, University of Ni\v{s}, Vi\v{s}egradska 33, 18\,000 Ni\v{s}, Serbia}

\emails{{\tt ravindra.bisht@yahoo.com}\,\,(R.\,K.\,Bisht), {\tt vrakoc@ptt.rs}\,\,(V.\,Rako\v cevi\' c)}

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\begin{abstract}
This paper points out several remarks on the paper of Pathak and
Rai
\begin{itemize}
\item[] H. K. Pathak  and  D. Rai, \emph{Common fixed point theorems for
PD-operator pairs under relaxed conditions with applications},
J. Comput. Appl. Math. \textbf{239}(2013), 103--113.
\end{itemize}
In fact, under contractive conditions (assumed in the
above paper), proving the existence of a common fixed point by assuming
the notion of a PD-operator is equivalent to proving the
existence of common fixed point by assuming the existence of a common
fixed point.
\end{abstract}

\keywords{coincidence point, point of coincidence, common fixed point}

\ams{54H25, 47H10}


\maketitle

\section{Introduction}

Pathak and Rai \cite{Pathak2013} define a PD-operator
pair of single valued mappings and obtain some common fixed point
theorems for this class of maps under relaxed conditions. Their
theorem generalizes results of Bhatt et al. \cite{Bhatt2010}, Jungck
and Rhoades \cite{Jungck2006}, Hussain et al. \cite{Hussain2011}
and others. As applications, they also study the solution of
functional equations arising from dynamic programming and
variational inequalities arising in the two point obstacle problem.

In 1982, Sessa \cite {Sessa1982} gave the first weaker version of
commutativity condition \cite{Jungck1976}, namely weakly commuting
mappings. In recent years  Jungck \cite{Jungck1986, Jungck1996},
Pant \cite{Pant1994}, Pathak et al. \cite{Pathak2013}, Al-Thangafi
and Shahzad \cite{Al-Thagafi2008} and many others \cite{Murthy2001}
have considered several generalizations of commuting mappings or
weaker notions of commutativity. Now, it has been shown that weak
compatibility is the minimal noncommuting condition for the
existence of common fixed points of contractive type mapping pairs.
In recent work, several authors have claimed to introduce some weaker
noncommuting notions and pretended to show weak compatibility as a
proper subclass of their weaker notions. This is, however, not true.
In view of the results of Alghamdi et al. (\cite{Alghamdi2012} see
also, \cite{Doric2012,Pant2012}), most of the generalized
commutativity notions fall into the subclass of weak compatibility in
the setting of a unique common fixed point (or unique point of
coincidence). If there are just two maps involved, and they only
have one coincidence point (which turns out to be the unique fixed
point), then of course a PD-operator pair and all of the
generalizations of commutativity coincide. If there are no
coincidence points, then there cannot be any fixed points and
PD-operator points either. Those generalizations of commuting
mappings are novel but for their actual applications one should go
beyond contractive conditions since contractive conditions do not
allow more than one point of coincidence or fixed point. In fact,
under contractive conditions proving the existence of common fixed
points by assuming several weaker noncommuting notions is often
equivalent to proving
the existence of common fixed points by assuming the existence of common
fixed points \cite{Pant2012}.

For completeness of the paper and convenience of the reader, in
Section 2 we collect some basic definitions and facts which are
applied in subsequent sections.

\section{Preliminaries}

Following Pathak and Rai \cite{Pathak2013} (see also
\cite{Alghamdi2012}) we assume $(X, d)$ denotes a metric space, and
for $x \in X$ and $A \subset X,$ set $d(x, A) = \inf\{d(x, y) : y
\in A\}.$

 Let $f$ and $g$ be self mappings
of a set $X$. If $w=f x =gx $ for some $x$ in $X$, then $x$ is
called a coincidence point of $f$ and $g$, and $w$ is called a point
of coincidence $(POC)$ of $f$ and $g$. The set of coincidence points
of $f$ and  $g$ will be denoted by $C(f, g)$. Let $PC (f, g)$
represent the set of points of coincidence of $f$ and $g$. A point
$x \in X$ is a common fixed point of $f$ and $g$ if $x=fx = gx $.
The set of all common fixed points of $f$ and $g$
is denoted by $F(f, g)$.

\begin{definition} A pair of  selfmaps $(f,g)$  of a
metric space $(X,d)$ is said to be
\begin{itemize}

\item[$($i$)$] commuting \cite {Jungck1976}, if $fgx=gfx$ for all $x$ in $X$;

\item[$($ii$)$] weakly commuting \cite {Sessa1982}, if $d(fgx, gfx)\leq d(fx,gx)$ for all $x$ in $X$;

\item[$($iii$)$] $R-$weakly commuting \cite {Pant1994}, if $d(fgx, gfx)\leq Rd(fx,gx)$ for all $x$ in $X$ and $R>0$;

\item[$($iv$)$] compatible \cite {Jungck1986}, if and only if  $\lim_{n}d(fgx_{n},gfx_{n})=0$,
whenever $\left\{x_{n}\right\}$ is a sequence in $X$ such that $\lim_{n}fx_{n}=\lim_{n}gx_{n}=t$ for some $t$ in $X$;

\item[$($v$)$] weakly compatible (WC) \cite{Jungck1996}, if the
pair commutes on the set of coincidence points, i.e., $fgx=gfx$
whenever
$fx=gx$ for some $x\in X$;

\item[$($vi$)$] occasionally weakly compatible (OWC)
\cite{Al-Thagafi2008}, if there
exists a coincidence point $x$ in $X$ such that $fx=gx$ implies $fgx=gfx$;

\item[$($vii$)$] a $PD$-operator pair \cite{Pathak2013}, if there
is a point $x \in  X$ such that $x \in C(f , g)$ and
\[
d(fgx, gfx) \leq diam(PC(f , g)), \text{ for some } x\in C(f ,g).
\]
\end{itemize}
\end{definition}

\begin{definition}
Let $X$ be a non-empty set and $d$ a
function $d : X \times X \rightarrow [0,\infty)$ such that
\begin{eqnarray} \label{eq1.1}
d(x, y)=0 \textrm{ if and only if } x=y, \textrm{ for each }x, y \in X.
\end{eqnarray}
\end{definition}

For a space $(X, d)$ satisfying \eqref{eq1.1} and $A \subset X$, the diameter of $A$ is defined by
\[
diam(A) = \sup\{\max\{d(x, y), d(y, x)\} : x, y \in A\}.
\]
Let us recall the following recent result.

\begin{proposition} [see \cite{Doric2012}]
Let a pair of mappings $(f, g)$ have a unique POC.
Then it is WC if and
only if it is OWC.
\end{proposition}

\section{Main results}

We start with an auxiliary result.

\begin{proposition} \label{proposition2}
Let $d : X \times X \rightarrow [0,\infty)$ be a
mapping such that $d(x, y)= 0$ if and only if $x = y.$
Let a pair of mappings $(f, g)$ have a unique POC. If it is a pair of $PD$-operators,
then it is WC.
\end{proposition}

\begin{proof} First we have that $C(f,g)\neq
\phi$ because $PC(f,g)\neq \phi$ ($PC (f, g)$ is a singleton). Since
$(f, g)$ is a $PD$-operator, then there exists some $x\in C(f,g)$
such that $d(fgx,gfx) \leq diam(PC(f, g))=0$. Hence $d(fgx,gfx)=0,$
i.e., there exists $fx=gx$ with $fgx=gfx$. Therefore the pair
$(f,g)$ is OWC. According to \cite {Doric2012}, $(f,g)$ is WC.
\end{proof}

\begin{proposition}
Let $\phi : R_+\rightarrow  R_+$ be a nondecreasing function
satisfying the condition $\phi(t) < t,$ for each $t > 0$, and let
$d: X\times X \rightarrow [0,\infty)$ be a mapping such that
$d(x,y)= 0$ if and only if $x = y.$ Suppose  that $(f,g)$ is a
$PD$-operator pair and satisfying the following condition:
\begin{eqnarray} \label{eq1.2}
d(fx, fy) \leq \phi (max\{d(gx, gy), d(gx, fy),
d(fx, gy), d(gy, fy)\}),
\end{eqnarray}
for each $x, y \in  X$. Then $f$ and $g$ are WC.
\end{proposition}

\begin{proof}
By hypothesis, there exists some
$x \in X$ such that $w= fx =gx.$ It remains to show that $(f, g)$
has a unique POC. Suppose there exists another point $w_1~=~fy =~gy$
with $w\neq w_1$. Then, we have
\begin{eqnarray*}
d(w,w_1)&=&d(fx, fy) \\
&\leq &\phi (max\{d(gx, gy), d(gx, fy), d(fx, gy), d(gy,fy)\})\\
&< &d(w,w_1),
\end{eqnarray*}
a contradiction. Thus $(f, g)$ has a unique POC. By Proposition~\ref{proposition2},
the pair $(f, g)$ is WC.
\end{proof}


In a recent paper, Pathak and Rai \cite{Pathak2013} proved the following theorem:

\begin{theorem}[see \cite{Pathak2013}]\label{theorem1}
Let $X$ be a nonempty set
and $d : X \times X \rightarrow [0,\infty)$ a function satisfying
condition \eqref{eq1.1}. Suppose that the pair
$(f,g)$ is a $PD$-operator satisfying condition \eqref{eq1.2}, then $(f,g)$ have a
unique common fixed point.
\end{theorem}

Now we prove our main result.
\begin{theorem}\label{theorem2}
Under contractive condition \eqref{eq1.2}
assumed in Theorem~\ref{theorem1}, the assumption of $PD$-operators and the existence
of a unique common fixed point are equivalent conditions.
\end{theorem}

\begin{proof}
We first observe that under contractive
condition \eqref{eq1.2}, the assumption of  $PD$-operators and the existence of
a unique common fixed point are equivalent conditions. To see this,
first suppose that $f$ and $g$ satisfy the contractive condition
\eqref{eq1.2}. If $f$ and $g$ have a common fixed point, say $z,$ then
$z=fz=gz,$ $fgz=gfz=z$. Thus,  $f$ and $g$ are $PD$-operators, since
contractive condition \eqref{eq1.2} excludes the existence of two points of
coincidence or common fixed points.

On the other hand, suppose that  $f$ and $g$ are  $PD$-operators
such that $fx=gx$ and
\[
d(fgx, gfx) \leq diam(PC(f,g))
\]
for some $x \in C(f,g)$. Now,  in view of condition \eqref{eq1.2}, we get
$diam(PC(f,g))=0,$ (since contractive condition \eqref{eq1.2} excludes the
existence of two points of coincidence). Hence $fgx=gfx$. If $fx\neq ffx$, using \eqref{eq1.2} we get
\begin{eqnarray*}
d(ffx, fx) &\leq& \phi(\max(d(gfx, gx), d(gfx, fx), d(ffx, gx), d(gx, fx)))\\
&<& d(ffx, fx),
\end{eqnarray*}
a contradiction. Hence $fx=ffx$ and
$fx=ffx=gfx$.  This means that $fx$ is a common fixed point of $f$
and $g$. Uniqueness of the fixed point follows from  contractive condition \eqref{eq1.2}.
\end{proof}


\begin{remark}
Let us remark that the  same results in
Theorem~\ref{theorem2}  will also be true for many contractive
conditions assumed in paper \cite{Pathak2013}, e.g.,
\begin{eqnarray}\label{eq1.3}
d(fx, fy) < \max(d(gx, gy), d(gx, fy), d(gy, fx), d(gy, fy)).
\end{eqnarray}

Therefore, under contractive conditions \eqref{eq1.2} and \eqref{eq1.3} the existence of
a common fixed point and $PD$-operators are equivalent conditions.
In order to find actual applications of  $PD$-operators one should
go beyond contractive conditions, since contractive conditions do
not allow more than one point of coincidence
or fixed point.
\end{remark}

\begin{problem} It would be interesting to know if the related results of
Theorem~\ref{theorem2} are true in nonself cases (\cite{Gajic2005,Gajic2007})?
\end{problem}


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\end{thebibliography}
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