In this paper it is proved the following main result that if $T$ is a self-map on a complete metric space $X, \rho$ and if there exists an upper semicontinuous bounded above function $G: X \to R$ such that $\rho [x,Tx] \leq G(Tx)-G(x)$ for every $x \in X$, then $T$ has a fixed point in $X$. This paper presents and some other results of this type.