Document Type : Research Paper
Author
School of Mathematics and Computer Sciences, Damghan University, P.O. Box 36715-364 Damghan, Iran
Abstract
Inspired by the work of Suzuki in
[T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness,
Proc. Amer. Math. Soc. 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,
Proc. Amer. Math. Soc., 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, Nonlinear Anal., 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.
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