Fixed-Point Properties of Roughly Contractive Mappings

Abstract

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

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 $2r_{\mathrm{conv}S}(S)/\mathrm{diam}S$ over all non-empty, non-singleton and bounded subsets $S$ of some normed linear space $X$, where $r_{\mathrm{conv}S}(S)={\inf }_{x\in \mathrm{conv}S}{\sup }_{y\in S}\Vert x-y\Vert$ is the self-radius of $S$ and $\mathrm{diam}S$ 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 $\epsilon >0$ there exists $x^{\ast }\in M$ such that

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 $\Vert x^{\ast }-T{x}^{\ast }\Vert \le \frac{1}{2}{J}_s(X)r$.