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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 $\varphi: \bbfR^+\to \bbfR^+$ satisfying $\varphi(0)=0$, $\varphi(s) <s$ for $s>0$ and that $\varphi$ is right upper semicontinuous such that $d(Tx,Ty) \le\varphi (d(x,y))$ $\forall x,y\in X$. {\it D. W. Boyd} and {\it J. S. W. Wong} [Proc. Am. Math. Soc. 20, 458-464 (1969; Zbl 0175.44903)] showed that $T$ has a unique fixed point. Later, {\it A. Meir} and {\it 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: $$\forall\varepsilon >0\ \exists \delta>0 \text{ such that }\varepsilon\le d(x,y)< \varepsilon+ \delta\Rightarrow d(Tx,Ty)< \varepsilon \tag 1$$ 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 {\it 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 $d(x,y)$.

54H25Fixed-point and coincidence theorems in topological spaces
54E50Complete metric spaces
54E40Special maps on metric spaces
Full Text: DOI
[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, No. 5, 26-42 (1972) · Zbl 0249.54026
[4] Wong, C. S.: Characterizations of certain maps of contractive type. Pacific J. Math. 68, No. 1, 293-296 (1977) · Zbl 0357.54022