The author presents some fixed point theorems for a (compact convex)-valued multimap \(T\) in a Banach space \(X\) satisfying the following condition: if \(A\) is a countable subset of \(X\), then \(A \subseteq \overline {\text{conv}} (C\cup T(A))\) for a given relatively compact set \(C \subset X\) implies that \(A\) is relatively compact. Reviewer's remark: In the reviewer's opinion, the correctness of the main results of the paper is not quite clear, since the compactness of the set \(C^*\) is not well justified in the proof of Theorem 2.1.