Supernormal cones and fixed point theory
Let (E,Spec(E)) be a locally convex space as defined by Treves. We denote by E' the topological dual of E. A subset \(K\subset E\) is said to be a convex cone if (i) \(K+K\subset K\), (ii) \(\forall \lambda \in R_+\), \(\lambda\) \(K\subset K\). If K is a convex cone, we denote by K' the dual cone of K with respect to the duality \(<E,E'>\). Let \(\tau\) be the locally convex topology defined on E by the set Spec(E). We say that a convex cone \(K\subset E\) is \(\tau\)-supernormal iff there exists a basis B of Spec(E) such that \(\forall p\in B\), \(\exists f_ p\in K'\) such that \(\forall x\in K\), \(p(x)\leq f_ p(x)\). The main theorem of this paper reads as follow: Theorem. Let \((E,Spec(E)=\{p_{\alpha}\}_{\alpha \in A})\) be a locally convex space ordered by a supernormal cone \(K\subset E\), and let \(S\subset E\) be a subset. A mapping \(f: S\to E\) has a fixed point in S iff there exists a complete subset \(S_ 0\subset S\) and a continuous mapping \(\phi\) : \(S\to E\) such that (i) \(f(S_ 0)\subset S_ 0\), (ii) \(\phi (S_ 0)\) is bounded, (iii) \(\phi\) (f(x))\(\geq \phi (x)\), \(\forall x\in S_ 0\); and (iv) \(\forall \alpha \in A\), \(\exists \beta \in A\) and \(c_{\beta}>0\) such that \(\forall x\in S_ 0\); \[ p_{\alpha}(x- f(x))\leq c_{\beta}p_{\beta}(\phi (f(x))-\phi (x)). \]
scientific article; zbMATH DE number 4030316