# Fixed-Point Properties of Roughly Contractive Mappings

• ### Hoàng Xuân Phú

Institute of Mathematics, Hanoi, Vietnam ## 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 and , a self-mapping is said to be -roughly -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 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 of some normed linear space , where $$r_{\conv S}(S) = \inf_{x\in\conv S} \sup_{y\in S} \'x - y\'$$ is the self-radius of and $$\diam S$$ is its diameter. If is a closed and convex subset of some finite-dimensional normed space and if is -roughly -contractive, then for all $$\e > 0$$ there exists such that

$\'x^* - Tx^*\' < {\t{1\over 2}}\,J_s(X)\, r + \e.$

If , or is some two-dimensional strictly convex normed space, or is some Euclidean space, then there is satisfying $$\'x^* - Tx^*\' \le {1 \over 2}\,J_s(X)\,r$$.