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Fixed point theorems for singlevalued mappings and multivalued mappings on complete metric spaces. (English) Zbl 0768.54032
The first theorem of this paper is a simultaneous extension of Theorem 5 of [N. Mizoguchi and W. Takahashi, J. Math. Anal. Appl. 141, 177–188 (1989; Zbl 0688.54028)] and of Theorem 1 of [N. A. Assad and W. A. Kirk, Pac. J. Math. 43, 553–562 (1972; Zbl 0239.54032)]. It is concerned with set-valued contractive mappings and is related to a problem raised by the reviewer [Atti Accad. naz. Lincei, VIII. Ser., Rend., Cl. Sci. Fis. Mat. Natur. 57(1974), 194–198 (1975; Zbl 0329.47019)].
The second theorem extends a theorem of D. W. Boyd and J. S. W. Wong [Proc. Am. Math. Soc. 20, 458–464 (1969; Zbl 0175.44903)]. Variants of Caristi’s fixed point theorem [J. Caristi, Trans. Am. Math. Soc. 215, 241–251 (1976; Zbl 0305.47029)] and of Ekeland’s variational principle [I. Ekeland, J. Math. Anal. Appl. 47, 324–353 (1974; Zbl 0286.49015)] are also included.

54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems
54C60 Set-valued maps in general topology