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

Document Type: Research Paper

Author

School of Mathematics and Computer Sciences‎, ‎Damghan University‎, ‎P.O‎. ‎Box 36715-364‎ ‎Damghan‎, ‎Iran

Abstract

‎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 $T:X\to X$‎, ‎the condition‎
‎\[‎
‎x,y\in X,\ x\neq y,\ d(x,Tx) \leq d(x,y)\ \Longrightarrow\‎
‎d(Tx,Ty) < \phi(d(x,y))‎,
‎\]‎
‎implies that the iterations $x_n=T^nx$‎, ‎for any choice of initial point $x\in X$‎, ‎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.

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