New fixed point theorems

for a family of mappings and applications

to problems on sets with convex sections

Authors:
Kunquan Lan and Jeffrey Webb

Journal:
Proc. Amer. Math. Soc. **126** (1998), 1127-1132

MSC (1991):
Primary 46A55, 47H04, 47H10, 54H25; Secondary 49J40

DOI:
https://doi.org/10.1090/S0002-9939-98-04347-0

MathSciNet review:
1451816

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Abstract | References | Similar Articles | Additional Information

Abstract: Some new fixed point theorems for a family of mappings are obtained and applied to problems on sets with convex sections that were first studied by Ky Fan.

**[Br]**F.E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces,*Math.Ann.***177**(1968), 283-301. MR**37:4679****[Fan]**Ky Fan, Sur un théorème minimax,*C.R.Acad.Sci. Paris*,**259**(1964), 3925-3928. MR**30:5145****[Fan1]**Ky Fan, Applications of a theorem concerning sets with convex sections,*Math. Ann.***163**(1966), 189-203. MR**32:8101****[Fan2]**Ky Fan, Some properties of convex sets related to fixed point theorems,*Math. Ann.***266**(1984), 519-537. MR**85i:47060****[Fan3]**Ky Fan, Fixed point and minimax theorems in locally convex linear spaces,*Proc. Nat. Acad. Sci. U.S.A.***38**(1952), 121-126. MR**13:858d****[Lan]**K. Q. Lan, A generalization of -space and some results on multivalued mappings without convexity, J. Math. Anal. Appl.**194**(1995), 511-528. MR**96f:54054****[Ma]**T.W. Ma, On sets with convex sections,*J. Math. Anal. Appl.***27**(1969), 413-416. MR**40:1676****[S.T]**M.H. Shih and K.K. Tan, Non-compact sets with convex sections,*Pacific J. Math.***119**(1985), 473-479. MR**87e:52006****[S.T.K]**M.H. Shih and K.K. Tan, Non-compact sets with convex sections, II,*J. Math. Anal. Appl.***120**(1986), 264-270. MR**87m:52004****[T]**E. Tarafdar, On minimax principles and sets with convex sections,*Publ. Math. Debrecen***29**(1982), 219-226. MR**84a:47074****[T.T]**E. Tarafdar and H.B. Thompson, On Ky Fan's minimax principle,*J. Aust. Math. Soc. Series A.***26**(1978), 220-226. MR**80a:47086****[Ty]**A. Tychonoff, Ein Fixpunktsatz,*Math Ann.***111**(1935), 767-776.**[Y.P]**N.C. Yannelis and N.D. Prabhakar, Existence of maximal elements and equilibria in linear topological spaces,*J. Math. Econom.***12**(1983), 233-245. MR**87h:90061a**

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Additional Information

**Kunquan Lan**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom

Email:
kl@maths.gla.ac.uk

**Jeffrey Webb**

Affiliation:
Department of Mathematics, University of Glasgow, Glasgow G12 8QW, United Kingdom

Email:
jrlw@maths.gla.ac.uk

DOI:
https://doi.org/10.1090/S0002-9939-98-04347-0

Keywords:
Fixed point,
partition of unity,
sets with convex sections.

Received by editor(s):
September 23, 1996

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society