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

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 $x\mathrm{\ne }y$, and, for every $ϵ>0$, there is $\delta >0$ such that $d(x,y)<\delta +ϵ$ implies $d(Tx,Ty)\le ϵ$. 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 $m(x,y)=d(x,y)+\gamma (d(x,Tx)+d(y,Ty))$, 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.