## 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].

### MSC:

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