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\title{On some qualitative properties of \'{C}iri\'{c}'s fixed point theorem}
\author{M\u{a}d\u{a}lina Moga}
\address{Babe\c{s}-Bolyai University, \\ Faculty of Mathematics and Computer Sciences\\
1, Kog\u{a}lniceanu Street,\\
400084 Cluj-Napoca,\\
Romania}
\email{madalina.moga@math.ubbcluj.ro}
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\subjclass{47H10; 54H25}
\keywords{metric space, fixed point, \'{C}iri\'{c} type operator, graphic contraction, data dependence, Ostrowski stability, Ulam-Hyers stability, well-posedness.}
\begin{abstract}
It is well known that of all the extensions of the Banach-Caccioppoli Contraction Principle, the most general result was established by \'{C}iri\'{c} in 1974.
In this paper, we will present some results related to \'{C}iri\'{c} type operator in complete metric spaces. Existence and uniqueness are re-called and several stability properties (data dependence and Ostrowski stability property) are proved. Using the retraction-displacement condition, we will establish the well-posedness and the Ulam-Hyers stability property of the fixed point equation $x=f(x)$.
\end{abstract}
\maketitle

\section{Introduction and Preliminaries}

																								   
																										 

Let $(X,d)$ be a  metric space and $f:X \rightarrow X$ be an operator.
For $A\subset X$, let $\delta(A):=sup\left \{ d(a,b): a,b\in A \right \}$ the diameter of the set $A$. 
For each $x\in X$, we denote: 
\begin{equation*}
    O(x,n)=\left \{ x, f(x),..., f^n(x) \right \}, n = 1, 2,...,
\end{equation*}
\begin{equation*}
    O(x,\infty)=\left \{ x, f(x),...,  f^n(x),...\right \}.
\end{equation*}

\begin{definition}{(\'{C}iri\'{c} \cite{C:1974})}
Let $(X,d)$ be a metric space and $f:X \rightarrow X$ be an operator. Then $X$ is said to be $f$-orbitally complete if every Cauchy sequence which is contained in $O(x,\infty)$, for some $x\in X$ converges in $X$.
\end{definition}

The following classes of operators in a metric space $(X,d)$ are important for our approach.

\begin{definition}{}
Let $(X,d)$ be a metric space and $f:X \rightarrow X$ be an operator. Then $f$ is said to be an $\alpha$-contraction if there exists $\alpha \in[0,1)$ such that
\begin{equation}
    d(f(x),f(y))\leq \alpha d(x,y), \mbox{ for all } x,y\in X.
\end{equation}

\end{definition}

\begin{definition}{(Rus \cite{R:2016})}
Let $(X,d)$ be a metric space and $f:X \rightarrow X$ be an operator. Then $f$ is said to be a graphic $\alpha$-contraction if there exists $\alpha \in[0,1)$ such that
\begin{equation}
    d(f^2(x),f(x))\leq \alpha d(x,f(x)), \mbox{ for all } x\in X.
\end{equation}
				 
\end{definition}

Through this paper we denote $\mathbb{N}:=\{0,1,2,\cdots \}$ the set of all natural numbers and by $\mathbb{N}^*=\mathbb{N}\setminus\{0\}$.

We recall that, $Fix(f)=\left \{ x\in X |x=f(x) \right \}$ is the fixed point set of $f$ and we denote by  $(f^n(x))_{n\in\mathbb{N}}$ the sequence of Picard iterates for $f$ starting from $x_0\in X$, where $f^n=f\circ f\circ\cdots\circ f$ for $n$-times. Notice that the sequence of Picard iterates for $f$ starting from $x_0\in X$ can be recursively defined by the formula $x_{n+1}=f(x_n), 
\text{ for } n\in \mathbb{N}$, where  $x_n:=f^n(x_0), n\in \mathbb{N}.$

\begin{definition}{(\'{C}iri\'{c} \cite{C:1974})}
An operator $f:X \rightarrow X$ is said to be a generalized contraction if and only if for every $x,y\in X$ there exists nonnegative numbers q, r, s and t, 
which may depend on both $x$ and $y$, such that $sup\left \{ q+r+s+2t: x,y\in X \right \}<1$ and
\begin{equation*}
\aligned
    \quad d(f(x),f(y))&\leq q\cdot d(x,y)+r\cdot d(x,f(x))+ \\ 
    & + s\cdot d(y,f(y))+t\cdot \left [ d(x,f(y))+d(y,f(x)) \right ].
\endaligned
\end{equation*}
\end{definition}

\begin{definition}{(\'{C}iri\'{c} \cite{C:1974})}
Let $(X,d)$ be a metric space and $f:X \rightarrow X$ be an operator. Then $X$ is said to be a \'{C}iri\'{c} type operator (named a quasi-contraction in the original paper \cite{C:1974}) 
if there exists a number $q \in (0,1)$, such that 
\begin{equation}
    d(f(x),f(y))\leq q\cdot max \left \{ d(x,y), d(x,f(x)), d(y,f(y)), d(x,f(y)), d(y,f(x)) \right \},
\end{equation}
for all $x,y\in X$.
\end{definition}

									   
																																																													   
							
														 
			  
												 
				 
													 
			   
				

It is well known (see \cite{Rh}) that of all the extensions of Banach-Caccioppoli Contraction Principle, the most general result was established by \'{C}iri\'{c} in 1974 for the above class of operators.
				 
																	  
			   

In following example  we present a \'{C}iri\'{c} type operator which is not a generalized contraction.
				 
																		  
			   
												
				 
														   
			   

																						   
\begin{example}
Let
\begin{equation*}
   X_1 = \left \{ \frac{m}{n}: m = 0, 1, 2, 4, 6, ...; n = 1, 3, 7, ..., 2k+1, ... \right \}, 
\end{equation*}
\begin{equation*}
    X_2 =  \left \{ \frac{n}{n}: m = 1, 2, 4, 6, 8,...; n = 2, 5, 8, ..., 3k+2, ... \right \},
\end{equation*}

where $k \in \mathbb{N}$ and let $X = X_1 \cup X_2$. Let us define $f:X \rightarrow X$ by
\begin{equation*}
    f(x)=\left\{\begin{matrix}
\dfrac{2}{3}x &, x\in X_1, \\ \\
\dfrac{1}{5}x &, x\in X_2. 
\end{matrix}\right.
\end{equation*}

The mapping $f$ is a \'{C}iri\'{c} type operator  with $q=\dfrac{2}{3}$. If both $x$ and $y$ are in $X_1$ or in $X_2$, then 
\begin{equation*}
    d(f(x),f(y))\leq \frac{2}{3} d(x,y)
\end{equation*}

If we take $x\in X_1$ and $y\in X_2$, then we have that
\begin{equation*}
    x \geq \frac{3}{10} y \text{ implies } d(f(x),f(y)) = \dfrac{2}{3} \left ( x-\frac{3}{10}y \right )\leq \dfrac{2}{3}\left ( x-\dfrac{1}{5}y \right ) = \dfrac{2}{3} d(x, f(y))
\end{equation*}

\begin{equation*}
    x< \frac{3}{10} y \text{ implies } d(f(x),f(y)) = \frac{2}{3}\left ( \frac{3}{10} y-x \right ) \leq \frac{2}{3}(y-x) = \frac{2}{3} d(x,y)
\end{equation*}


Thus, we have that $f$ satisfies the following condition:
\begin{equation*}
    d(f(x),f(y))\leq \frac{2}{3} max\left \{ d(x, y), d(x, f(y)), d(y, f(x)) \right \}
\end{equation*}
and, hence, it is \'{C}iri\'{c} type operator.

In the following step we show that $f$ is not a generalized contraction on $X$. Let $x = 1$ and $y = \frac{1}{2}$. Then we have that

\begin{equation*}
\aligned
   \quad q\cdot d(x,y)&+r\cdot d(x,f(x))+s \cdot d(y,f(y))+t\cdot \left [ d(x,f(y))+d(y,f(x)) \right ] = \\
   & = \frac{1}{2}q + \frac{1}{3}r+ \frac{4}{10}s + \frac{32}{30}t < (q+r+s+2t) \frac{32}{60} < \\
   & < \frac{32}{60} < \frac{17}{30} = d(f(x),f(y)),
   \endaligned
\end{equation*}
as $q+r+s+2t < 1$, we can see that $f$ is not a generalized contraction.

\end{example}

In this paper, we will present some results related to \'{C}iri\'{c} type operator in complete metric spaces. Existence and uniqueness are re-called and several stability properties (data dependence and Ostrowski stability property) are proved. Using the retraction-displacement condition, we will establish the well-posedness and the Ulam-Hyers stability property of the fixed point equation $x=f(x)$.

Our results generalize and complement some theorems given in \cite{BMR}, \cite{C:1974}, \cite {P:2014}, \cite{R:2001}, \cite{R:2016}, \cite{IAR}, \cite{R:2008}.

\section{Main Results}

In this section we will consider a metric space $(X,d)$ and $f:X\rightarrow X$ a \'{C}iri\'{c} type operator. Besides the usual properties which are proved by \'{C}iri\'{c} in \cite{C:1974}, 
we will prove some other stability properties. More precisely. we will establish the continuous data dependence property of the fixed point and the Ostrowski stability property for the operator $f$. Moreover, using the retraction-displacement condition and we also prove that the fixed point equation $x=f(x)$ is well-posed and Ulam-Hyers stable.

\begin{theorem}{(\'{C}iri\'{c} \cite{C:1974})}\label{CiricTh}
Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be a \'{C}iri\'{c} type operator. Suppose that $X$ is $f$-orbitally complete. Then:
\begin{enumerate}
													
    \item $f$ has a unique fixed point $x^*$ in $X$ and $\displaystyle \lim_{n\rightarrow \infty} f^n(x)=x^*$, i.e., $f$ is a Picard operator;
    \item $d(f^n(x),x^*)\leq \frac{q^n}{1-q}d(x,f(x))$, for every $x\in X$ and every $n\in \mathbb{N}^*$;
\end{enumerate}
\end{theorem}

The idea of the proof is based on the following two relations: 

\quad (i)  if $n\in \mathbb{N}^*$, then for each $x\in X$ we have that $d(f^i(x),f^j(x))\le q \delta(O(x,n))$, for every $i,j\in \mathbb{N}^*$;

\quad (ii) for each $x\in X$ we have that $\delta( O(x,\infty))\le \frac{1}{1-q} d(x,f(x))$. 

A second result in \cite{C:1974} shows that if there exists $p\in \mathbb{N}$ with $p\ge 2$
such that $f^p$ is a  \'{C}iri\'{c} type operator, then $f$ is a Picard operator.

\begin{remark}
    If $f:X\rightarrow X$ satisfies all the assumptions in Theorem \ref{CiricTh}, then we have the following additional conclusion:

3. $f$ satisfies the retraction-displacement condition 
\begin{equation}\label{DR}
d(x,x^*)\leq \frac{1}{1-q} d(x,f(x)), \mbox{ for all  } x\in X.
\end{equation}
\end{remark}

\begin{remark}
The conclusion 3. follows by 2. in the following way. Take $n=1$ in 2. Then,  we have 
\begin{equation*}
    d(f(x),x^*)\leq \frac{q}{1-q}d(x,f(x)), \text{ for all } x\in X.
\end{equation*}
Hence
\begin{equation*}
    d(x,x^*)\leq d(x,f(x))+d(f(x),x^*)\leq \frac{1}{1-q}d(x,f(x)), \text{ for all } x\in X.
\end{equation*}
\end{remark}

\begin{lemma}[Cauchy-Toeplitz Lemma]
    Let $(a_n)_{n\in \mathbb{N}}, (b_n)_{n\in \mathbb{N}}$ be two sequences of positive numbers such that $\displaystyle \sum_{n\geq 0} a_n < \infty$ and $\displaystyle \lim_{n\rightarrow \infty} b_n = 0$. Then 
    \begin{equation*}
        \displaystyle \lim_{n\rightarrow \infty}\left ( \sum_{k=0}^{n}
    a_{n-k}b_k \right )=0
    \end{equation*} 
\end{lemma}

The following notion is essential in our approach.

\begin{definition}{(Rus \cite{R:2008})}
Let $(X,d)$ be a metric space and $f:X\rightarrow X$ be an operator such that $Fix(f)\neq \emptyset$. 
We say that $f$ satisfies the retraction-displacement condition if there exists $c>0$ and a set retraction $\rho : X\rightarrow Fix(f)$ such that
\begin{equation}\label{Retr}
    d(x,\rho(x))\leq cd(x,f(x)), \text{ for all } x\in X.
\end{equation}
If $Fix(f) = \left \{ x^* \right \}$ then we have
\begin{equation*}
    d(x,x^*)\leq cd(x,f(x)), \text{ for all } x\in X.
\end{equation*}
\end{definition}

For example, if $f:X\rightarrow X$ is an $\alpha$-contraction and $(X,d)$ is a complete metric space then $f$ satisfies the following retraction-displacement condition
\begin{equation*}
    d(x,x^*)\leq \frac{1}{1-\alpha}d(x,f(x)), \text{ for all } x\in X.
\end{equation*}

On the same lines, if $f:X\rightarrow X$ is a graphic $\alpha$-contraction then it satisfies the retraction-displacement condition
\begin{equation*}
    d(x,\rho(x))\leq \frac{1}{1-\alpha}d(x,f(x)), \text{ for all } x\in X,
\end{equation*}
where $\rho :X \rightarrow Fix(f)$ is defined by
\begin{equation*}
    \rho(x)=\displaystyle \lim_{n\rightarrow\infty} f^n(x).
\end{equation*}





The following theorem is the main result of the paper.

\begin{theorem}
Let $(X,d)$ be a metric space, $f: X\rightarrow X$ be a \'{C}iri\'{c} type operator and suppose that $X$ is $f$-orbitally complete. 
Denote by $x^* \in X$ the unique fixed point of $f$. Then the following conclusions hold:
\begin{enumerate}
    \item the fixed point $x=f(x)$ equation has the data dependence property, i.e., for any operator $g:X\rightarrow X$ such that $Fix(g)\neq \emptyset$ and $d(f(x),g(x))\leq \eta$, for all $x\in X$ and some $\eta>0$, we have 
     \begin{equation*}
        d(x^*,u^*)\leq \dfrac{1+q}{1-q}\eta,
     \end{equation*}
    for all $u^*\in Fix(g)$.
    
    \item the fixed point equation is well-posed, i.e., for every sequence $ (u_n)_{n\in \mathbb{N}} \subset X$ such that $$d(u_n,f(u_n))\rightarrow 0,$$ as $n\rightarrow \infty$, we have that $u_n\rightarrow x^*$, as $n\rightarrow \infty$;
		
																																																																			 
	
		\item the fixed point equation is  Ulam-Hyers stable, i.e., there exists $c>0$ such that for any $\varepsilon >0$ and any $u^*\in X$ an $\varepsilon$-solution of the fixed point equation (in the sense that $d(u^*,f(u^*))\leq \varepsilon)$, we have 
    \begin{equation*}
        d(u^*,x^*)\leq c \cdot \varepsilon.
    \end{equation*}
    
    \item if $q<\frac{1}{2}$, then the fixed point equation has the Ostrowski stability pro-perty, i.e., for any sequence $\left ( u_n \right )_{n\in \mathbb{N}}\subset X$ with $d(u_{n+1},f(u_n))\rightarrow 0$ as $n\rightarrow \infty $, we have that $u_n\rightarrow x^*$;
    
\item if $q<\frac{1}{2}$, then $f$ is a graphic $\frac{q}{1-q}$-contraction;

\item if $q<\frac{1}{3}$, then the operator $f$ is a quasi-contraction, in the sense that there exists $\beta:=\frac{q}{1-2q}<1$ such that 
$$d(f(x),x^*)\le \beta d(x,x^*), \mbox{ for every } x\in X.$$ 
\end{enumerate}
\end{theorem}

\begin{proof}
\begin{enumerate}
\item To prove data dependence we will take $u^*\in Fix(g)$ such that $d(f(x),g(x))\leq\eta$. Then $d(x^*,u^*)\leq \dfrac{1+q}{1-q}\eta$.
\begin{equation*}
\aligned
\quad     d(x^*,u^*)&=d(f(x^*),g(u^*))\leq d(f(x^*),f(u^*))+d(f(u^*),g(u^*))\leq  \\
     & \leq q \cdot max\left \{ d(x^*,u^*),d(x^*,f(x^*)),d(u^*,f(u^*)),d(x^*,f(u^*)), \right. \\ 
     &\quad \left. \ d(y^*,f(u^*)) \right \}+ d(f(u^*),g(u^*)) \leq \\
     &\leq q \cdot max\left \{ d(x^*,u^*),d(u^*,g(u^*))+d(g(u^*),f(u^*)), \right.\\ 
     &\quad \left. d(x^*,g(u^*))+d(g(u^*),f(u^*)), d(x^*,u^*) \right \}+ \eta \leq \\
     & \leq q \cdot max \left \{ d(x^*,u^*), \eta, d(x^*,u^*)+\eta, d(x^*,u^*) \right \} + \eta \\
     & \leq q(d(x^*,u^*)+\eta) +\eta
\endaligned
\end{equation*}
Hence, we get that 
\begin{equation*}
    d(x^*,u^*)\leq \frac{1+q}{1-q}\eta
\end{equation*}

\item We will prove that the fixed point equation is well-posed. Let us estimate the distance between $u_n$ and $x^*$, where $(u_n)_n\in\mathbb{N}$ 
is a sequence in $X$ such that $d(u_n, f(u_n))\rightarrow 0$ as $n\rightarrow \infty$.

In order to prove this, we will use the retraction-displacement condition (\ref{DR}). We have:
\begin{equation*}
    d(u_n,x^*)\leq \frac{1}{1-q} d(u_n,f(u_n)) \rightarrow 0 \text{ as } n \rightarrow \infty
\end{equation*}

\item  Let $\varepsilon >0$ and $u^* \in X$ be an $\varepsilon$-solution of the fixed point equation $x=f(x)$, i.e., 
$d(u^*,f(u^*))\leq \varepsilon$. Using the retraction-displacement condition (\ref{DR}) we will estimate the distance between $x^*$ and $u^*$:
\begin{equation*}
    d(x^*,u^*)=d(u^*,x^*)\leq \frac{1}{1-q}d(u^*,f(u^*))\leq \frac{1}{1-q} \varepsilon 
\end{equation*}

There exists $c>0$ such that $c:=\frac{1}{1-q}$. 
Then it follows that
\begin{equation*}
    d(x^*,u^*)\leq c \cdot \varepsilon
\end{equation*}
which proves that the fixed point equation $x=f(x)$ is Ulam-Hyers stable.

\item We will show that the operator $f:X\rightarrow X$ has the Ostrowski property. We observe that:
    \begin{equation}\label{eqO1}
     d(u_{n+1},x^*)\leq d(u_{n+1},f(u_n))+d(f(u_n),x^*)
    \end{equation}
    We take separately $d(f(u_n),x^*)$ from the above inequality and we have:
    \begin{equation*}
    \aligned
    \quad d(f(u_n),x^*)&=d(f(u_n),f(x^*))\leq \\
    & \leq  q\cdot max \left \{ d(u_n,x^*), d(u_n,f(u_n)), d(x^*,f(x^*), d(u_n,f(x^*)), \right.\\
    & \quad \left. d(x^*,f(u_n)) \right \} \leq \\
    & \leq q\cdot max \left \{ d(u_n,x^*), d(u_n,x^*)+d(x^*,f(u_n)), d(u_n,x^*), \right.\\ 
    & \quad \left. d(x^*,f(u_n)) \right \}\leq \\
    & \leq q(d(u_n,x^*)+d(x^*,f(u_n))).
    \endaligned
    \end{equation*}
    Thus
    \begin{equation}\label{eqO2}
    d(f(u_n),x^*)\leq \frac{q}{1-q}d(u_n,x^*)
    \end{equation}
    We replace in \eqref{eqO1} the relation obtained in inequality \eqref{eqO2}:
    \begin{equation}
     d(u_{n+1},x^*)\leq d(u_{n+1},f(u_n))+\frac{q}{1-q} d(u_n,x^*)
    \end{equation}
    We denote: $\alpha:=\dfrac{q}{1-q}<1$.\\
    We will use Cauchy-Toeplitz Lemma and we obtain:
    \begin{equation*}
    \aligned
    \quad d(u_{n+1},x^*)&\leq d(u_{n+1},f(y_n))+\alpha d(u_n,x^*)\leq \\
    & \leq  d(u_{n+1},f(u_n))+\alpha [d(u_n,f(u_{n-1}))+\alpha d(u_{n-1},x^*)]\\
    & \leq d(u_{n+1},f(u_n))+\alpha d(u_n,f(u_{n-1}))+\alpha ^2d(u_{n-1},x^*)\leq ... \leq\\ 
    & \leq d(u_{n+1},f(u_n))+\alpha d(u_n,f(u_{n-1}))+\alpha ^2d(u_{n-1},f(u_{n-2}))+\\ 
    & \quad +...+\alpha^n d(u_1,f(u_0))+\alpha ^{n+1}d(u_0,x^*)\rightarrow 0 \text{ as } n\rightarrow \infty.
    \endaligned
    \end{equation*}
    
\item If we put $y:=f(x)$ in the \'{C}iri\'{c} type operator condition, we get 
$$d(f(x),f^2(x))\le q\max\{d(x,f(x)), d(f(x),f^2(x)), d(x,f^2(x)) \}\le$$ 
$$q\left(d(x,f(x))+d(f(x),f^2(x))\right).$$
Thus, we get that $d(f(x),f^2(x))\le \frac{q}{1-q} d(x,f(x))$, for every $x\in X$.

\item We will show now that $f$ is a quasi-contraction, in the sense that $$d(f(x),x^*)\le \beta d(x,x^*), \mbox{ for every } x\in X,$$
where $\beta:=\frac{q}{1-2q}<1$. Indeed, by the third conclusion of Theorem \ref{CiricTh} for $n=1$, we have
$d(f(x),x^*)\leq \frac{q}{1-q}d(x,f(x))$, for every $x\in X$. Then, we can write successively:
$$d(f(x),x^*)\leq \frac{q}{1-q}d(x,f(x))\le \frac{q}{1-q} \left(d(x,x^*)+d(f(x),x^*)\right).$$
As a consequence, $$d(f(x),x^*)\leq \frac{q}{1-2q} d(x,x^*), \mbox{ for each } x\in X.$$
																							

\end{enumerate}
\end{proof}




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\bibitem{R:2008} Rus, I.A., Petru\c{s}el, A., Petru\c{s}el, G., \emph{Fixed Point Theory}, Cluj Univ. Press, Cluj-Napoca, 2008.
\end{thebibliography}

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