Leray-Schauder type alternatives and the solvability of complementarity problems
The author gives existence theorems for nonlinear complementarity problems (NCP) in Hilbert spaces. The NCP(\(f,K\)) is defined for a convex cone \(K\) and a mapping \(f\) as to find some \(x^*\in K\) such that \(f(x^*)\in K\) and \((x^*,f(x^*))=0\). The proof of the results base on Leray-Schauder type alternatives. After introducing some basic notions and preliminary results it is proved the equivalence of the solvability of NCP\((f,K)\) and the absence of an exceptional family of \(f\) with respect to \(K\). The equivalence is stated under the assumption that \(f\) is a completely continuous field, an \(\alpha\)-condensing field or a projectionally pseudo-contractant field. In the rest of the paper different conditions ensuring the absence of an exceptional family are presented. The first condition given, which is called (\(\theta - S\)) condition, generalizes the (\(\theta\)) condition the author used in [\textit{G. Isac} and \textit{A. Carbone}, J. Glob. Optim. 15, No. 2, 181--196 (1999; Zbl 0957.90136)]. The author then extends to an infinite dimensional Hilbert space setting the conditions used in [\textit{P. Harker} and \textit{J. S. Pang}, Math. Program., Ser. B 48, No. 2, 161--220 (1990; Zbl 0734.90098)] and in [\textit{Y. B. Zhao} and \textit{J. Han}, J. Glob. Optim. 14, No. 3, 313--330 (1999; Zbl 0932.49012)]. Both conditions are proved to be implied by the (\(\theta - S\)) condition. Finally the author introduces the concept of a weakly proper mapping \(f\) on \(K\). This condition also generalizes a similar notion used in the above mentioned paper of Zhao and Han and is shown to be equivalent to the solvability of NCP(\(f,K\)) in case that \(f\) is a completely continuous field, an \(\alpha\)-condensing field or a projectionally pseudo-contractant field.
scientific article; zbMATH DE number 1751369