×

On characterizations of Meir-Keeler contractive maps. (English) Zbl 1009.54044

From the text: Let \((X,d)\) be a complete metric space and \(T:X \to X\) a map. Suppose there exists a function \(\varphi: \mathbb{R}^+\to \mathbb{R}^+\) satisfying \(\varphi(0)=0\), \(\varphi(s) <s\) for \(s>0\) and that \(\varphi\) is right upper semicontinuous such that \(d(Tx,Ty) \leq\varphi (d(x,y))\) \(\forall x,y\in X\). D. W. Boyd and J. S. W. Wong [Proc. Am. Math. Soc. 20, 458-464 (1969; Zbl 0175.44903)] showed that \(T\) has a unique fixed point. Later, A. Meir and E. Keeler [J. Math. Anal. Appl. 28, 326-329 (1969; Zbl 0194.44904)] extended Boyd-Wong’s result to mappings satisfying the following more general condition: \[ \forall\varepsilon >0\;\exists \delta>0 \text{ such that }\varepsilon\leq d(x,y)< \varepsilon+ \delta\Rightarrow d(Tx,Ty)< \varepsilon \tag{1} \] In this paper, we characterize condition (1) in terms of a \(\varphi\) function as in Boyd-Wong’s theorem. This is obviously desirable since then one can easily see how much more general is Meir-Keeler’s result than Boyd-Wong’s. A characterization was given earlier by C. S. Wong [Pac. J. Math. 68, 293-296 (1977; Zbl 0357.54022)], but it was in terms of a function \(\delta\) imposed on \(d(Tx,Ty)\) rather than \(d(x,y)\).

MSC:

54H25 Fixed-point and coincidence theorems (topological aspects)
54E50 Complete metric spaces
54E40 Special maps on metric spaces
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Boyd, D. W.; Wong, J. S.W., On nonlinear contractions, Proc. Am. Math. Soc., 20, 458-464 (1969) · Zbl 0175.44903
[2] Meir, A.; Keeler, E., A theorem on contraction mappings, J. Math. Anal. Appl., 28, 326-329 (1969) · Zbl 0194.44904
[3] Reich, S., Fixed points of contractive functions, Boll. Un. Mat. Ital., 4, 5, 26-42 (1972) · Zbl 0249.54026
[4] Wong, C. S., Characterizations of certain maps of contractive type, Pacific J. Math., 68, 1, 293-296 (1977) · Zbl 0357.54022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.