Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s. (English) Zbl 1137.54026

The author gives a very simple proof of Banach’s contraction principle for set-valued mappings with nonempty bounded closed values in a complete metric space due to N. Mizoguchi and W. Takahashi [J. Math. Anal. Appl. 141, No.1, 177–188 (1989; Zbl 0688.54028)]. It is illustrated by an example that this result is a real generalization of a fixed point theorem of S. B. Nadler Jr. [Pac. J. Math. 30, 475–488 (1969; Zbl 0187.45002)], in contrast to A. A. Eldred, J. Anuradha and P. Veeramani [J. Math. Anal. Appl. 336, No. 2, 751–757 (2007; Zbl 1128.47051)].


54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
Full Text: DOI


[1] Alesina, A.; Massa, S.; Roux, D., Punti uniti di multifunzioni con condizioni di tipo boyd – wong, Boll. unione mat. ital. (4), 8, 29-34, (1973) · Zbl 0274.54036
[2] Banach, S., Sur LES opérations dans LES ensembles abstraits et leur application aux équations intégrales, Fund. math., 3, 133-181, (1922) · JFM 48.0201.01
[3] Daffer, P.Z.; Kaneko, H., Fixed points of generalized contractive multi-valued mappings, J. math. anal. appl., 192, 655-666, (1995) · Zbl 0835.54028
[4] Daffer, P.Z.; Kaneko, H.; Li, W., On a conjecture of S. Reich, Proc. amer. math. soc., 124, 3159-3162, (1996) · Zbl 0866.47040
[5] Eldred, A.A.; Anuradha, J.; Veeramani, P., On equivalence of generalized multi-valued contractions and Nadler’s fixed point theorem, J. math. anal. appl., 336, 751-757, (2007) · Zbl 1128.47051
[6] Mizoguchi, N.; Takahashi, W., Fixed point theorems for multivalued mappings on complete metric spaces, J. math. anal. appl., 141, 177-188, (1989) · Zbl 0688.54028
[7] Jachymski, J., On Reich’s question concerning fixed points of multimaps, Boll. unione mat. ital. (7), 9, 453-460, (1995) · Zbl 0863.54042
[8] Nadler, S.B., Multi-valued contraction mappings, Pacific J. math., 30, 475-488, (1969) · Zbl 0187.45002
[9] Reich, S., Some problems and results in fixed point theory, Contemp. math., 21, 179-187, (1983) · Zbl 0531.47048
[10] Semenov, P.V., Fixed points of multivalued contractions, Funct. anal. appl., 36, 159-161, (2002) · Zbl 1026.47040
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.