We say that a nonlinear operator $N: E \rightarrow E $ on a Banach space $E$ is cone preserving (i.e. positive) if it maps the cone $K \subseteq E$ into itself, that is $N(K) \subseteq K$.
My question is: does $N(0)=0$? In other words, does $0$ map to $0$ under $N$? I know that if the operator is linear, then this is true. But I am not sure if it is still true for a general, nonlinear operator which is positive.
If $N$ preserves every cone, then it preserves $\{0\}$. Thus, $$N(0) \in N(\{0\}) \subseteq \{0\} \implies N(0) = 0.$$