Abstract :
Inspired by the work of Suzuki in
[T.~Suzuki, A generalized Banach contraction principle that characterizes metric completeness,
\textit{Proc. Amer. Math. Soc.}, \textbf{136} (2008), 1861--1869],
we prove a fixed point theorem for contractive mappings
that generalizes a theorem of Geraghty in [M.A. Geraghty, On contractive mappings,
\textit{Proc. Amer. Math. Soc.}, \textbf{40} (1973), 604--608]
and characterizes metric completeness. We introduce the family $\A$ of all nonnegative functions
$\phi$ with the property that, given a metric space $(X,d\,)$ and a mapping $T:X\to X$,
the condition
\[
x,y\in X,\ x\neq y,\ d(x,Tx) \leq d(x,y)\ \Longrightarrow\
d(Tx,Ty) < \phi(d(x,y)),
\]
implies that the iterations $x_n=T^nx$, for any choice of initial point $x\in X$,
form a Cauchy sequence in $X$. We show that the family of L-functions,
introduced by Lim in [T.C. Lim, On characterizations of Meir-Keeler contractive maps,
\textit{Nonlinear Anal.}, \textbf{46} (2001), 113--120], and the family
of test functions, introduced by Geraghty, belong to $\A$. We also prove
a Suzuki-type fixed point theorem for nonlinear contractions.
