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Weak univalence and connectedness of inverse images of continuous functions. (English) Zbl 0977.90060
Summary: A continuous function \(f\) with domain \(X\) and range \(f(X)\) in \(R^n\) is weakly univalent if there is a sequence of continuous one-to-one functions on \(X\) converging to \(f\) uniformly on bounded subsets of \(X\). In this article, we establish, under certain conditions, the connectedness of an inverse image \(f^{-1}(q)\). The univalence results of Radulescu-Radulescu, Moré-Rheinboldt, and Gale-Nikaido follow from our main result. We also show that the solution set of a nonlinear complementarity problem corresponding to a continuous \(P_0\)-function is connected if it contains a nonempty bounded clopen set; in particular, the problem will have a unique solution if it has a locally unique solution.

MSC:
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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