Let \(K\) be a finite union of closed convex subsets of a topological vector space \(E\). The set \(K\) is said to be pseudoconvex if there exists a finite-dimensional subspace \(E_0\) of \(E\) such that for any finite-dimensional subspace \(E'\) of \(E\) with \(E_0\subset E'\), the set \(K\cap E'\) is a retract of \(E'\).
The author proves that a compact admissible (in the sense of Górniewicz) map \(F:Y\multimap K\) with nonempty compact values has a fixed point in the case when \(Y,K\) are pseudoconvex subsets of a locally convex Hausdorff topological space satisfying \(Y\subset K\) and \(F(\partial_K Y)\subset Y\).
Next, the author introduces a new concept of measure of noncompactness and obtains a fixed point theorem for condensing admissible maps \(F:Y\multimap K\) with nonempty compact values (the sets \(Y,K\) been as above).
Finally, the author gives another fixed point theorem in locally convex spaces for quasicompact or ultimately compact admissible maps with nonempty compact values.