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Zeitschrift für Analysis und ihre Anwendungen


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Volume 22, Issue 3, 2003, pp. 517–528
DOI: 10.4171/ZAA/1159

Published online: 2003-09-30

Fixed-Point Properties of Roughly Contractive Mappings

Hoàng Xuân Phú[1]

(1) Institute of Mathematics, Hanoi, Vietnam

\def\s{\smallskip} \def\t{\textstyle} \def\R{{\Bbb R}} \def\conv{\mathop{\rm conv}} \def\diam{\mathop{\rm diam}} \def\ri{\mathop{\rm ri}} \def\e{\varepsilon} \def\ol{\overline} \def\wh{\widehat} \def\wt{\widetilde} For given $k \in (0,1)$ and $r > 0$, a self-mapping $T: \, M \to M$ is said to be $r$-roughly $k$-contractive provided $$ \'Tx - Ty\' \le k\,\'x - y\' + r \quad (x,y \in M). $$ To state fixed-point properties of such a mapping, the self-Jung constant $J_s(X)$ is used, which is defined as the supremum of the ratio $2\,r_{\conv S}(S)/\diam S$ over all non-empty, non-singleton and bounded subsets $S$ of some normed linear space $X$, where $r_{\conv S}(S) = \inf_{x\in\conv S} \sup_{y\in S} \'x - y\'$ is the self-radius of $S$ and $\diam S$ is its diameter. If $M$ is a closed and convex subset of some finite-dimensional normed space $X$ and if $T: \, M \to M$ is $r$-roughly $k$-contractive, then for all $\e > 0$ there exists $x^* \in M$ such that $$ \'x^* - Tx^*\' < {\t{1\over 2}}\,J_s(X)\, r + \e. $$ If $\dim X = 1$, or $X$ is some two-dimensional strictly convex normed space, or $X$ is some Euclidean space, then there is $x^* \in M$ satisfying $\'x^* - Tx^*\' \le {1 \over 2}\,J_s(X)\,r$.

Keywords: Roughly contractive mapping, fixed point theorem, rough invariance, self-Jung constant

Phú Hoàng Xuân: Fixed-Point Properties of Roughly Contractive Mappings. Z. Anal. Anwend. 22 (2003), 517-528. doi: 10.4171/ZAA/1159