Some variants of contraction principle, generalizations and applications
Dedicated to Professor Gheorghe Coman on the occasion of his 80th anniversary
Keywords:
Metric space, generalized metric space, contraction, quasicontraction, generalized contraction, Bessaga operator, Janos operator, Picard operator, - Picard operator, well-posedness of fixed point problem, Ostrowski property, data dependence, Ulam stability, iterative algorithm, iterative algorithm stability, stability under operator perturbation, functional differential equation, functional integral equation.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.
Mathematics Subject Classification (2010): 47H10, 54H25, 65J15, 34Kxx, 45Gxx, 45N05.
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