Let $\gamma_+$, $\gamma_-:\mathbb{R}_+\to\mathbb{R}$ be two given functions. Assume that $\gamma_+$ ($\gamma_-$) is smooth, strictly increasing (decreasing) and $\gamma_{+}(+\infty)=+\infty$ ($\gamma_{-}(+\infty)=-\infty$).

Now consider the following system:

\begin{eqnarray} \lambda''(\gamma_+)\gamma_+'\gamma_+&=&\lambda''(\gamma_-)\gamma_-'\gamma_-,~ \forall l>0 \\ \lambda'(\gamma_+)-\lambda'(\gamma_-)&=&2g'(l)+2\lambda''(\gamma_-)\gamma_-'\gamma_-,~ \forall l>0, \end{eqnarray}

wherer $\gamma_+=\gamma_+(l)$, $\gamma_-=\gamma_-(l)$ and $g=g(l)$ is a given function regular enough. Now my question is how to obtain explicitly the solution $\lambda: \mathbb{R}\to\mathbb{R}$ and under which conditions $\lambda$ is convex. Thx a lot for the reply!