On a conjecture of S. Reich

Authors:
Peter Z. Daffer, Hideaki Kaneko and Wu Li

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3159-3162

MSC (1991):
Primary 47H10

MathSciNet review:
1363454

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Abstract: Simeon Reich (1974) proved that the fixed point theorem for single-valued mappings proved by Boyd and Wong can be generalized to multivalued mappings which map points into compact sets. He then asked (1983) whether his theorem can be extended to multivalued mappings whose range consists of bounded closed sets. In this note, we provide an affirmative answer for a certain subclass of Boyd-Wong contractive mappings.

**1.**D. W. Boyd and J. S. W. Wong,*On nonlinear contractions*, Proc. Amer. Math. Soc.**20**(1969), 458–464. MR**0239559**, 10.1090/S0002-9939-1969-0239559-9**2.**T. H. Chang, Common Fixed Point Theorems for Multi-Valued Mappings,Math. Japonica 41 (1995), 311--320. CMP**1995:11****3.**P. Z. Daffer and H. Kaneko, Fixed Points of Generalized Contractive Multi-Valued Mappings,J. Math. Anal. Appl. 192 (1995), 655--666. CMP**1995:12****4.**Noriko Mizoguchi and Wataru Takahashi,*Fixed point theorems for multivalued mappings on complete metric spaces*, J. Math. Anal. Appl.**141**(1989), no. 1, 177–188. MR**1004592**, 10.1016/0022-247X(89)90214-X**5.**Simeon Reich,*Fixed points of contractive functions*, Boll. Un. Mat. Ital. (4)**5**(1972), 26–42 (English, with Italian summary). MR**0309095****6.**Simeon Reich,*Some fixed point problems*, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8)**57**(1974), no. 3-4, 194–198 (1975) (English, with Italian summary). MR**0397487****7.**Simeon Reich,*Some problems and results in fixed point theory*, Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), Contemp. Math., vol. 21, Amer. Math. Soc., Providence, RI, 1983, pp. 179–187. MR**729515**, 10.1090/conm/021/729515**8.**K. P. R. Sastry, S. V. R. Naidu, and J. R. Prasad,*Common fixed points for multimaps in a metric space*, Nonlinear Anal.**13**(1989), no. 3, 221–229. MR**986445**, 10.1016/0362-546X(89)90051-5

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Additional Information

**Peter Z. Daffer**

Affiliation:
Department of Mathematics, Macon College, Macon, Georgia 31297

Email:
pdaffer@cennet.mc.peachnet.edu

**Hideaki Kaneko**

Affiliation:
Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529-0077

Email:
kaneko@math.odu.edu

**Wu Li**

Affiliation:
Department of Mathematics and Statistics, Old Dominion University, Norfolk, Virginia 23529-0077

Email:
wuli@math.odu.edu

DOI:
https://doi.org/10.1090/S0002-9939-96-03659-3

Received by editor(s):
April 5, 1995

Communicated by:
Dale Alspach

Article copyright:
© Copyright 1996
American Mathematical Society