In this paper it is proved that if $T$ is a self-map on a complete metric space $(X,\rho)$ and if there exist a lower semicontinuous function $G:\to \mathbb{R}_{+}^{0}$ and an arbitrary fixed integer $k\geq 0$ such that $\rho[x,Tx]\leq G(x)-G(Tx)+\cdots +G(T^{2k}x)-G(T^{2k+1}x)$ and $G(T^{2i+1}x)\leq G(T^{2i}x)$ for $i=0,1,\ldots,k$ and for every $x\in X$, then $T$ has a fixed point $\xi$ in $X$.