Fixed-Point Properties of Roughly Contractive Mappings

Abstract

\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 $x^{\ast }\in M$ 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 $x^{\ast }\in M$ satisfying \( \'x^* - Tx^*\' \le {1 \over 2}\,J_s(X)\,r \).