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 ‎ϕ with the property that‎, ‎given a metric space (X,d) and a mapping T:X→X ‎, ‎the condition‎ ‎‎x,y∈X, x≠y, d(x,Tx)≤d(x,y) ⟹\‎‎d(Tx,Ty)<ϕ(d(x,y))‎,‎ ‎implies that the iterations x n =T n x ‎, ‎for any choice of initial point x∈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.