In the paper, a partial answer to a conjecture formulated by \textit{R. D. Nussbaum} [Trans. Am. Math. Soc. 171, 349-375 (1972; Zbl 0256.47040)], regarding an asymptotic version of the Schauder fixed point theorem, is given. The result (Theorem 3) may be read as follows: Let \(M\) be a nonempty convex and closed subset of a Banach space \(X\) and \(T:M \to M\) a continuous mapping. Suppose \(T\) is proper and there exists an integer \(n \geq 2\) such that \(T^n\) is compact. Then \(T\) has a fixed point.