Boundary Value Problems for Nonlinear ODEs

At the beginning of Chap. 4, we observed that to find radial solutions of (P≥) and (P≤) one is lead to solve the following ODE: $$\displaystyle \big [v_g\varphi (w')\big ]' = v_g \beta f(w) l(|w'|) $$ on an interval of \(\mathbb R^+_0\), where we have extended φ to an odd function on all of \(\mathbb R\). The functions vg and β are bounds, respectively, for the volume of geodesic spheres of M and for b.