  We introduce the notion of admissible functions and show that the family of
L-functions introduced by Lim in [Nonlinear Anal. 46(2001), 113--120] and the
family of test functions introduced by Geraghty in [Proc. Amer. Math. Soc.,
40(1973), 604--608] are admissible. Then we prove that if $\phi$ is an
admissible function, $(X,d)$ is a complete metric space, and $T$ is a mapping
on $X$ such that, for $\alpha(s)=\phi(s)/s$, the condition
$1/(1+\alpha(d(x,Tx))) d(x,Tx) < d(x,y)$ implies $d(Tx,Ty) < \phi(d(x,y))$, for
all $x,y\in X$, then $T$ has a unique fixed point. We also show that our fixed
point theorem characterizes the metric completeness of $X$.
