JournalszaaVol. 22 , No. 3DOI 10.4171/zaa/1159

Fixed-Point Properties of Roughly Contractive Mappings

  • Hoàng Xuân Phú

    Institute of Mathematics, Hanoi, Vietnam
Fixed-Point Properties of Roughly Contractive Mappings cover

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(0,1)k \in (0,1) and r>0r > 0, a self-mapping T:MMT: \, M \to M is said to be rr-roughly kk-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 Js(X)J_s(X) is used, which is defined as the supremum of the ratio 2r\convS(S)/\diamS2\,r_{\conv S}(S)/\diam S over all non-empty, non-singleton and bounded subsets SS of some normed linear space XX, where r_{\conv S}(S) = \inf_{x\in\conv S} \sup_{y\in S} \'x - y\' is the self-radius of SS and \diamS\diam S is its diameter. If MM is a closed and convex subset of some finite-dimensional normed space XX and if T:MMT: \, M \to M is rr-roughly kk-contractive, then for all \e>0\e > 0 there exists xMx^* \in M such that

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

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