On a Problem of Krasnosel'skii and Rutickii

In \cite[p. 30]{5}, M. A. Krasnosel'skii and Ya. B. Rutickii proposed a problem, which can be reformulated as follows. Let $f$ be an $N$-function such that $f(ts)\leq f(t)f(s)$, $s,t\geq 1$. Is there another $N$-function $F$ such that $F(st)\leq F(t)F(s)$, $s,t>0$ and equivalent to $f$ on $[1,\infty)?$. We give a necessary and sufficient condition for a positive and constructive solution.