Positivity 8: 315–326, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Fixed Point Theorems on Product Topological
Spaces and Applications
XIE PING DING
, JONG YEOUL PARK
and IL HYO JUNG
Department of Mathematics, Sichuan Normal University, Chengdu, Sichuan 610066, P.R. China;
Department of Mathematics, Pusan National University, Kumjung, Pusan 609-735, Korea
(Received 6 June 2001; accepted 1 July 2002)
Abstract. A new collectively ﬁxed point theorem for a family of set-valued mappings deﬁned on
product spaces of non-compact topological spaces without linear structure is proved and some special
cases are also discussed. As applications, some non-empty intersection theorems of sets with convex
sections and equilibrium existence theorem of abstract economies are obtained under much weaker
assumptions. Our results includes a number of known results as many special cases.
Key words: collectively ﬁxed point; product space; contractible; sets with convex section; abstract
Recently, Tarafdar  ﬁrst established a collectively ﬁxed point theorem for a
family of set-valued mappings deﬁned on the product space of compact convex
subsets of topological vector spaces and gave some applications to mathematical
economies, game theory and the problems of social sciences. By using a partition
of unity and Tychonoff’s ﬁxed point theorem, Lan and Webb , Ansari and Yao
 and Singh, Tarafdar and Watson  have obtained some generalizations of
Tarafdar’s ﬁxed point theorem in  where the domain and range sets of mappings
may not be compact. Some applications to the section theorem of Ky Fan type,
equilibrium existence problems of abstract economies and system of variational
inequalities were also given.
In this paper, we shall prove a new collectively ﬁxed point theorem for a family
of set-valued mappings deﬁned on the product spaces of general non-compact to-
pological spaces without linear structure under much weaker assumptions. Our new
theorem includes all collectively ﬁxed point theorems mentioned above and many
known ﬁxed point theorems of Fan-Browder type as special cases, As applications,
some new non-empty intersection theorems of sets with convex sections and equi-
The ﬁrst author was supported by the NNSF of China (19871059), the KOSEF of Korea and the
NSF of Education Department of Sichuan Province of China( 25).
The second and third authors were supported by the Korean research foundation, No. 1998-15-