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.