On some boundary value problems for second order nonlinear differential equations. (Q2918589)

The authors investigate the second-order differential equation NEWLINE\[NEWLINE(a| x'| ^{p-2}x')' = bF(x),NEWLINE\]NEWLINE where \(a,b>0\), \(uF(u)>0\) for \(u\neq 0\) and \(\lim_{u\to 0} \frac {F(u)}{| u| ^{p-2} u} \in [0,\infty )\). Sufficient conditions for the existence of a positive solution \(x\) in \([0,\infty )\) tending to zero for \(t\to \infty \) are given; estimates for \(x\) are also derived. A comparison theorem for principal solutions of two equations of the above given type with different \(b\) is proved.