A singular initial value problem for the equation ${\displaystyle u^{\left(n\right)}\left(x\right)=g\left(u\left(x\right)\right)}$

We consider the problem of the existence of positive solutions u to the problem ${\displaystyle u^{\left(n\right)}\left(x\right)=g\left(u\left(x\right)\right)}$, ${\displaystyle u\left(0\right)=u\prime \left(0\right)=...=u^{(n-1)}\left(0\right)=0}$ (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition ${\displaystyle \int {₀}^{\delta }\frac{1}{s}{\left[\frac{s}{g\left(s\right)}\right]}^{\frac{1}{n}}ds<\infty }$ is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.