Suzuki-type fixed point theorems for generalized contractive mappings‎ ‎that characterize metric completeness

‎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 $T:Xto X$‎, ‎the condition‎ ‎[‎ ‎x,yin X, xneq 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 $xin X$‎, ‎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.