Fixed point theorems for the class \(Q(X,Y)\) (Q2575940)

This article deals with set-valued maps \(T: X \to Y\) of a new class \(Q(X,Y)\), where \(X\) is a subset of a Hausdorff topological vector space \(E\), \(Y\) a Hausdorff topological vector space. This class is defined as a set of maps with the following property: for any compact convex subset \(K\) of \(X\) and any continuous function \(f: T(K) \to K\), the composition \(F(T| _K) \to 2^K\) has a fixed point. Subclasses of \(Q(X,Y)\) are the class of continuous functions, the class \(K(X,Y)\) of Kakutani maps (with convex values and codomains being convex spaces), the class of acyclic maps \(V(X,Y)\), the class of approachable maps, and some others. Furthermore, the authors consider nearly convex subsets \(X \subseteq E\); namely, a subset \(X\) is called nearly convex if for every compact subset \(A \subseteq X\) and every neighbourhood \(V\) of the origin \(0\), there is a continuous mapping \(h: A \to X\) such that \(x - h(x) \in V\) for all \(x \in A\) and \(h(A)\) is contained in some convex subset of \(X\). The main results are five fixed point theorems. In particular (Theorems 2.1 and 2.4), the authors prove that \(T \in Q(X,X)\) has a fixed point in \(X\) either if \(T\) is closed and \(\overline{TX} \subseteq X\) is compact, or if \(T\) is a closed \(\Phi\)-condensing map (\(\Phi\) is a measure of noncompactness with standard properties).