Maximal element principles on generalized convex spaces and their applications. (English) Zbl 1018.47036

Agarwal, Ravi P. (ed.) et al., Set valued mappings with applications in nonlinear analysis. London: Taylor & Francis. Ser. Math. Anal. Appl. 4, 149-174 (2002).
The author defines a new class of set-valued mappings on \(G\)-convex spaces and derives several coincidence and KKM-theorems for these mappings. Consider a class \(\mathcal{U}\) of set-valued mappings satisfying the following properties: (i) \(\mathcal{U}\) contains all continuous single-valued mappings, (ii) any finite composition of mappings in \(\mathcal{U}\) is upper semicontinuous with nonempty compact values, and (iii) every finite composition of mappings in \(\mathcal{U}\) mapping the standard \(n\)-simplex into itself has a fixed point. Given such a class \(\mathcal{U}\), the author considers the set \(\mathcal{U}^\kappa\) of all mappings \(F\) such that for any compact set \(K\) in the domain of definition of \(F\) there is a mapping \(F^*\) on \(K\) belonging to \(\mathcal{U}^\kappa\) such that \(F^*(x)\subset F(x)\) whenever \(x\in K\). A typical result then reads as follows: Let \((X,D,\Gamma)\) be a \(G\)-convex space and \(Y\) a topological space. Let \(F:X\to 2^Y\) belong to \(\mathcal{U}^\kappa\). Let \(G:X\to 2^Y\) be a mapping such that for any nonempty finite subset \(N\) of \(D\) we have that \(F(\Gamma(N))\cap G^{-1}(\{x\})\) is relatively open in \(F(\Gamma(N))\) whenever \(x\in N\) and that \(F(\Gamma(N))\subset\bigcup_{x\in N}(Y\setminus G^{-1}(\{x\}))\). The conclusion is that for each nonempty finite subset \(N\) of \(D\) we have that \(F(\Gamma(N))\cap\left(\bigcap_{x\in N}(Y\setminus G^{-1}(\{x\}))\right)\not=\emptyset\) and that \(G(y)\cap N=\emptyset\) for some \(y\in F(\Gamma(N))\).
For the entire collection see [Zbl 0996.00018].


47H04 Set-valued operators
91B50 General equilibrium theory
47H10 Fixed-point theorems