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

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]**Felix E. Browder,*The fixed point theory of multi-valued mappings in topological vector spaces*, Math. Ann.**177**(1968), 283–301. MR**0229101****[Fan]**Ky Fan,*Sur un théorème minimax*, C. R. Acad. Sci. Paris**259**(1964), 3925–3928 (French). MR**0174955****[Fan1]**Ky Fan,*Applications of a theorem concerning sets with convex sections*, Math. Ann.**163**(1966), 189–203. MR**0190689****[Fan2]**Ky Fan,*Some properties of convex sets related to fixed point theorems*, Math. Ann.**266**(1984), no. 4, 519–537. MR**735533**, 10.1007/BF01458545**[Fan3]**Ky. Fan,*Fixed-point and minimax theorems in locally convex topological linear spaces*, Proc. Nat. Acad. Sci. U. S. A.**38**(1952), 121–126. MR**0047317****[Lan]**Kun Quan Lan,*A generalization of 𝐻-space and some results on multivalued mappings without convexity*, J. Math. Anal. Appl.**194**(1995), no. 2, 511–528. MR**1345052**, 10.1006/jmaa.1995.1314**[Ma]**È. B. Lebedeva and S. N. Šimanov,*The stability of quasiharmonic systems with retardation slowly changing in the time*, Izv. Vysš. Učebn. Zaved. Matematika**1968**(1968), no. 12 (79), 53–61 (Russian). MR**0248424****[S.T]**Mau-Hsiang Shih and Kok-Keong Tan,*Noncompact sets with convex sections*, Pacific J. Math.**119**(1985), no. 2, 473–479. MR**803130****[S.T.K]**Mau Hsiang Shih and Kok-Keong Tan,*Noncompact sets with convex sections. II*, J. Math. Anal. Appl.**120**(1986), no. 1, 264–270. MR**861918**, 10.1016/0022-247X(86)90214-3**[T]**E. Tarafdar,*On minimax principles and sets with convex sections*, Publ. Math. Debrecen**29**(1982), no. 3-4, 219–226. MR**678897****[T.T]**E. Tarafdar and H. B. Thompson,*On Ky Fan’s minimax principle*, J. Austral. Math. Soc. Ser. A**26**(1978), no. 2, 220–226. MR**511606****[Ty]**A. Tychonoff, Ein Fixpunktsatz,*Math Ann.***111**(1935), 767-776.**[Y.P]**Nicholas C. Yannelis and N. D. Prabhakar,*Existence of maximal elements and equilibria in linear topological spaces*, J. Math. Econom.**12**(1983), no. 3, 233–245. MR**743037**, 10.1016/0304-4068(83)90041-1

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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