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 $\phi$ is an admissible function, $(X,d)$ is a complete metric space, and $T$ is a mapping on $X$ such that, for $\alpha(s)=\phi(s)/s$, the condition $1/(1+\alpha(d(x,Tx))) d(x,Tx) < d(x,y)$ implies $d(Tx,Ty) < \phi(d(x,y))$, 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 $X$.

💡 Research Summary

The paper introduces a new class of functions called admissible functions and shows that two previously studied families—the L‑functions of Lim (2001) and the test functions of Geraghty (1973)—fit into this class. An admissible function φ is defined on