Let \(X\) be a real Banach space. A nonempty closed convex subset \(P\) of \(X\) is a cone if it is invariant under homothetic maps and does not contain any line segment through the origin. It is well known that \(P\) induces a partial order “\(\leq\)”on \(X\). A cone is said to be solid if it has nonempty interior. The interior of \(P\) is denoted by \(P^0\). A cone \(P\) is called normal if there exists \(\delta > 0\) such that \(\| x \| \leq \delta \| y \|\) whenever \(0 \leq x \leq y\). Let \(P\) be a cone in \(X\) and let \(0 \neq u_0 \in P\). Let \(T:X \rightarrow X\) be such that \(T(P) \subseteq P\). Then \(T\) is said to be {\(u_0\)-positive (relative to \(P\))} if for each \(0 \neq x \in P\) there exist \(\alpha(x) >0\), \(\beta(x) >0\) and \(n(x) \in \mathbb{N}\) such that \(\alpha(x) u_0 \leq T^{n(x)}x \leq \beta(x) u_0\). Now let \(P\) be solid. Then \(T\) (satisfying \(T(P) \subseteq P\)) is said to be strongly positive if for \(0 \neq x \in P\), there exists \(n(x) \in \mathbb{N}\) such that \(T^{n(x)}x \in P^0\). The authors prove that, if \(P\) is a normal solid cone in \(X\) and if \(T(P) \subseteq P\), then \(T\) is \(u_0\)-positive for some \(u_0 \in P^0\) if and only if \(T\) is \(u_0\)-positive for every \(u_0 \in P^0\), which in turn is necessary and sufficient for \(T\) to be strongly positive.