Hoang Xuan Phu and Tran Van Truong
 
Abstract
In this paper,  r-roughly k-contractive mappings T: M ---> M (which satisfy $d(T x, Ty) \leq kd(x,y)+r$ for all $x,y\in M$ and some $k\in (0,1)$, $r>0$) are considered. If M  is not assumed to be convex, T  is only guaranteed to admit $\gamma$-invariant points $x^*$ (which fulfill $d(x^*,Tx^*)\leq \gamma$) with $\gamma\geq r/(1-k)$. For M as a  compact convex subset, T  possesses $\gamma$-invariant points for all $\gamma>r$. If M is a closed  and convex subset of some normed space $(\Bbb R^n,|| . ||)$, then, for all $\varepsilon>0$, there exist  $\gamma$-invariant points with $\gamma= nr/(n+1) +\varepsilon$. If  the normed space $(\Bbb R^n,|| . ||)$ is strictly convex, then T  admits $\gamma$-invariant points with $\gamma= nr/(n+1)$. In particular, if  || . || is the Euclidean norm, then there are  $\gamma$-invariant points with $\gamma= (n/2(n+1))^{1/2} r$.