Intersection theorems with applications in optimization
The classical Fan-KKM lemma gives a nonempty intersection result expressed by a KKM-mapping \(S\): \(\bigcap_{x\in X} \not= \emptyset\). Originally, a KKM-mapping is a set-valued function \(S: X \twoheadrightarrow Y\), \(X\), \(Y\) nonempty subsets of a vector space, \(X\) convex, if for every nonempty finite subset \(A\) of \(X\) \[ \operatorname{conv}(A) \subset \bigcup_{x \in A}S(x).\tag{1} \] The point of the paper is, that the authors use an interesting generalization of that KKM-mapping (following [\textit{G. Botelho} et al., Rev. Mat. Complut. 23, No. 1, 139--161 (2010; Zbl 1210.46031)] and especially [\textit{G. Botelho} et al., Rev. Mat. Complut. 23, No. 1, 139--161 (2010; Zbl 1210.46031)]) and prove two new intersection theorems. Then they apply them for problems in optimization and variational analysis. The mentioned generalization is, that a second set-valued mapping \(T:X \twoheadrightarrow Y\) is used such that (1) is replaced by \[ T(\operatorname{conv}(A)) \subset \bigcup_{x \in A}S(x),\tag{2} \] or replaced by: for each subset \(A\) as above and any \(x \in \operatorname{conv}(A)\), \[ T(x) \cap S(A) \not= \emptyset. \tag{3} \] (1) is called KKM-mapping w.r.t. \(T\), (2) is called weak KKM-mapping w.r.t. \(T\). Both new intersection theorems give the nonempty intersection that there is \(x_0\in X\) such that \(T(x_0 \cap \bigcap_{u\in X}S(u) \not= \emptyset\) under the conditions that \(X, Y\) are sets in topological vector spaces and, in the first theorem, \(X\) nonempty, convex, compact, \(Y\) nonempty, convex, \(S\) a closed set-valued mapping with convex values and convex cofibers and weak-KKM w.r.t. \(T\), \(T\) has compact convex values and, finally, for each \(u\in X\), the set \(\{x\in X: T(x)\cap S(u)\not=\emptyset\}\) is closed. The second intersection theorem is a noncompact version of the first one and also with weak-KKM \(S\). The content-rich applications show that the skillful use of \(T\) in addition to \(S\) leads to improved solution conditions. The applications concern Stampaccchia variational inequalities, complementarity problems, different minimax inequalities, saddle point problems, and some fixed point problems. Having a look to complementarity problems in general topological vector spaces: Find \(x_0 \in X\) and \(x_0^* \in P(x_0)\) such that \(\langle x_0^*,x_0 \rangle = 0\) and \(\forall x\in X: \langle x_0^*,x \rangle \geq 0 \), where \(E\) a topological vector space, \(E^*\) its dual, \(X\) a nonempty subset of \(E\) and \(P: X \twoheadrightarrow E^* \) with nonempty values. Then it is easy to prove the existence of a solution, but the drawback is that in those proofs norm compactness is used in the dual space. But taking skillfully (under suitable conditions) weak-KKMS, weak* compactness can be used in \(E^*\) (where \(E\) is a Banach space) and the duality pairing is continuous (where it is used).
scientific article; zbMATH DE number 6985493