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Equivalent conditions and the Meir-Keeler type theorems. (English) Zbl 0834.54025
This is an important contribution to fixed point theory. The author shows among others that the contractive condition given in [{\it G. Jungck}, {\it K. B. Moon}, {\it S. Park}, and {\it B. E. Rhoades}, J. Math. Anal. Appl. 180, No. 1, 221-222 (1993; Zbl 0790.54055)] is equivalent to the contractive condition considered in [{\it S. Sessa}, {\it R. N. Mukherjee}, and {\it T. Som}, Math. Jap. 31, 235-245 (1986; Zbl 0603.54044)]. Moreover, he gives some new Meir-Keeler type conditions (ensuring convergence of successive approximations) which are equivalent to those given by the reviewer [Čas. Pěstování Mat. 105, 341-344 (1980; Zbl 0446.54042)] (in this paper there are some misprints in the statement of the main theorem). From the point of view of possible applications in the theory of functional equations, especially interesting is Theorem 2 which, in particular, unifies the Boyd-Wong theorem and a result of the reviewer [Diss. Math. 127 (1975; Zbl 0318.39005)] which is not included in the Meir-Keeler theorem. Some results for locally contractive mappings and $c$-chainable metric spaces are also given.

54H25Fixed-point and coincidence theorems in topological spaces
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