Some variants of contraction principle, generalizations and applications

Abstract

In this paper we present the following variant of contraction principle:

\noindent\underline{Saturated principle of contraction}. Let $(X,d)$ be a complete metric space and $f:X\to X$ be an $l$-contraction. Then we have:
\begin{itemize}
\item [$(i)$] $F_{f^n}=\{x^*\}$, $\forall\ n\in\mathbb{N}^*$.
\item [$(ii)$] $f^n(x)\to x^*$ as $n\to\infty$, $\forall\ x\in X$.
\item [$(iii)$] $d(x,x^*)\leq\psi(d(x,f(x)))$, $\forall\ x\in X$ where $\psi(t)=\frac{t}{1-l}$, $t\geq 0$.
\item [$(iv)$] $y_n\in X$, $d(y_n,f(y_n))\to 0$ as $n\to\infty$ $\Rightarrow $ $y_n\to x^*$ as $n\to\infty$.
\item [$(v)$] $y_n\in X$, $d(y_{n+1},f(y_n))\to 0$ as $n\to\infty$ $\Rightarrow $ $y_n\to x^*$ as $n\to\infty$.
\item [$(vi)$] If $Y\subset X$ is a nonempty bounded and closed subset with $f(Y)\subset Y$, then $x^*\in Y$ and $\displaystyle\bigcap_{n\in\mathbb{N}}f^n(Y)=\{x^*\}$.
\end{itemize}

The basic problem is: which other metric conditions imply the conclusions of this variant ?

We give some answers for this problem. Some applications and open problems are also presented.