# Fixed-Point Properties of Roughly Contractive Mappings

### Hoàng Xuân Phú

Institute of Mathematics, Hanoi, Vietnam

## Abstract

For given $k∈(0,1)$ and $r>0$, a self-mapping $T:M→M$ 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 $2r_{convS}(S)/diamS$ over all non-empty, non-singleton and bounded subsets $S$ of some normed linear space $X$, where $r_{convS}(S)=f_{x∈convS}sup_{y∈S}∥x−y∥$ is the self-radius of $S$ and $diamS$ is its diameter. If $M$ is a closed and convex subset of some finite-dimensional normed space $X$ and if $T:M→M$ is $r$-roughly $k$-contractive, then for all $ε>0$ there exists $x_{∗}∈M$ such that

If $dimX=1$, or $X$ is some two-dimensional strictly convex normed space, or $X$ is some Euclidean space, then there is $x_{∗}∈M$ satisfying $∥x_{∗}−Tx_{∗}∥≤21 J_{s}(X)r$.

## Cite this article

Hoàng Xuân Phú, Fixed-Point Properties of Roughly Contractive Mappings. Z. Anal. Anwend. 22 (2003), no. 3, pp. 517–528

DOI 10.4171/ZAA/1159