Let x'=f(t,x) be a differential equation with $f$ in the hypothesis of Picard's theorem. Let $\varphi$ be a solution such that its interval of definition contains $(t_0,+\infty)$ for some fixed $t_0\in \mathbb{R}$. Suppose $\lim_{t\to+\infty} \varphi(t)=a \in \mathbb{R}$. Must the constant function $t\mapsto a$ be a solution of the equation?
(This is a question I've asked myself while studying the logistic model x'=ax(1-x) and wondering how to justify the solutions look the way they do.)
EDIT: By "the hypothesis of Picard's theorem", I mean $f$ continuous and locally Lipschitz with respect to $x$.
I'm interested also in particular cases (e.g. autonomous system) where this holds.