I'm concerned with the discussion of a ODE, especially the discussion of the solution. I've got the assumptions that there is the relation $\varphi''(x)=F(\varphi(x))$ for all $x$ on $\mathbb{R}$. Forthermore I know (by the notation $\lim\limits_{x\rightarrow\infty} \varphi(x)=:\varphi_0$) that $F(\varphi_0)=0$ and $F'(\varphi_0)>0$. The statement now is: $\varphi(x)\rightarrow\varphi_0$ exponentially as $x\rightarrow\infty$ (and therefore $F'(\varphi(x))\rightarrow F'(\varphi_0)$ exponentially as well).
But I don't know how to check that statement. Maybe someone have got some ideas?