Fixed point theorems of Ciric-Matkowski type in generalized metric spaces

Abstract: A self-map $T$ of a $\nu$-generalized metric space $(X,d\,)$ is said to be a Ciric-Matkowski contraction if $d(Tx,Ty)<d(x,y)$, for $x\neq y$, and, for every $\epsilon\>0$, there is $\delta>0$ such that $d(x,y)<\delta+\epsilon$ implies $d(Tx,Ty)\leq \epsilon$. In this paper, fixed point theorems for this kind of contractions of $\nu$-generalized metric spaces, are presented. Then, by replacing the distance function $d(x,y)$ with functions of the form $m(x,y)=d(x,y)+\gamma\bigl(d(x,Tx)+d(y,Ty)\bigr)$, where $\gamma>0$, results analogue to those due to P.D. Proiniv (Fixed point theorems in metric spaces, Nonlinear Anal. 46 (2006) 546--557) are obtained.