From the text: Let be a complete metric space and a map. Suppose there exists a function satisfying , for and that is right upper semicontinuous such that . D. W. Boyd and J. S. W. Wong [Proc. Am. Math. Soc. 20, 458-464 (1969; Zbl 0175.44903)] showed that 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:
In this paper, we characterize condition (1) in terms of a 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 imposed on rather than .