A generalization to multifunctions of Fan's best approximation theorem

Authors:
V. M. Sehgal and S. P. Singh

Journal:
Proc. Amer. Math. Soc. **102** (1988), 534-537

MSC:
Primary 47H10

MathSciNet review:
928974

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a theorem for set valued mappings in an approximatively compact, convex subset of a locally convex space, and then derive results due to Ky Fan and S. Reich as corollaries.

**[1]**Ky Fan,*Extensions of two fixed point theorems of F. E. Browder*, Math. Z.**112**(1969), 234–240. MR**0251603****[2]**C. J. Himmelberg,*Fixed points of compact multifunctions*, J. Math. Anal. Appl.**38**(1972), 205–207. MR**0303368****[3]**J. L. Kelley and Isaac Namioka,*Linear topological spaces*, With the collaboration of W. F. Donoghue, Jr., Kenneth R. Lucas, B. J. Pettis, Ebbe Thue Poulsen, G. Baley Price, Wendy Robertson, W. R. Scott, Kennan T. Smith. The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J., 1963. MR**0166578****[4]**Tzu Chu Lin,*A note on a theorem of Ky Fan*, Canad. Math. Bull.**22**(1979), no. 4, 513–515. MR**563767**, 10.4153/CMB-1979-067-x**[5]**Simeon Reich,*Approximate selections, best approximations, fixed points, and invariant sets*, J. Math. Anal. Appl.**62**(1978), no. 1, 104–113. MR**0514991****[6]**V. M. Sehgal and S. P. Singh,*A theorem on the minimization of a condensing multifunction and fixed points*, J. Math. Anal. Appl.**107**(1985), no. 1, 96–102. MR**786014**, 10.1016/0022-247X(85)90355-5**[7]**V. M. Sehgal and S. P. Singh,*On random approximations and a random fixed point theorem for set valued mappings*, Proc. Amer. Math. Soc.**95**(1985), no. 1, 91–94. MR**796453**, 10.1090/S0002-9939-1985-0796453-1**[8]**V. M. Sehgal,*A simple proof of a theorem of Ky Fan*, Proc. Amer. Math. Soc.**63**(1977), no. 2, 368–369. MR**0435954**, 10.1090/S0002-9939-1977-0435954-8**[9]**C. Waters, Ph.D. thesis, University of Wyoming, 1984.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47H10

Retrieve articles in all journals with MSC: 47H10

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1988-0928974-5

Keywords:
Approximately compact set,
multivalued mappings,
upper and lower semicontinuous mappings,
condensing map,
uniformly convex

Article copyright:
© Copyright 1988
American Mathematical Society