A well known result of Browder states that if C is a nonempty compact and convex subset in \(R^ n\) and f:C\(\times [0,1]\to C\) is a continuous mapping, then there is a closed and connected subset T of \(C\times [0,1]\) which meets both \(C\times \{0\}\) and \(C\times \{1\}\) such that \(x\in f(x,t)\) for every (x,t) in T.
In the last decade there have been several attempts to generalize this important theorem. R. Saigal [Math. Oper. Res. 1, 260-266 (1976; Zbl 0363.90091)] extended it to multivalued mappings from \(R^ n\) to \(R^ n\) with compact convex values and A. Mas-Colell [Math. Program. 6, 229-233 (1974; Zbl 0285.90068)] proved that Saigal’s theorem remains true even for mappings with compact and contractible values.
One important application of Browder’s theorem is in nonlinear complementarity problems. B. C. Eaves [Math. Program. 1, 68-75 (1971; Zbl 0227.90044)] used it to prove the basic theorem of complementarity. Saigal [loc. cit.] used Mas-Colell’s extension of Browder’s theorem to extend Eaves’ result to multi-valued mappings with compact and contractible values.
In the first section of the present paper we shall extend Saigal’s fixed point theorem to an arbitrary locally convex space. In the second section this result is used to extend Eaves’ basic theorem of complementarity to barrel spaces. Finally, as an application we shall give an existence theorem for the generalized complementarity problem.