A Suzuki-type fixed point theorem for nonlinear contractions

We introduce the notion of admissible functions and show that the family of L-functions introduced by Lim in [Nonlinear Anal. 46(2001), 113–120] and the family of test functions introduced by Geraghty in [Proc. Amer. Math. Soc., 40(1973), 604–608] are admissible. Then we prove that if $\varphi$ is an admissible function, $(X,d)$ is a complete metric space, and $T$ is a mapping on $X$ such that, for $\alpha (s)=\varphi (s)\mathrm{/}s$, the condition $1\mathrm{/}(1+\alpha (d(x,Tx)))d(x,Tx)<d(x,y)$ implies $d(Tx,Ty)<\varphi (d(x,y))$, for all $x,y\in X$, then $T$ has a unique fixed point. We also show that our fixed point theorem characterizes the metric completeness of $X$.