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On some boundary value problems for second order nonlinear differential equations. (English) Zbl 1265.34113
The authors investigate the second-order differential equation \[ (a| x'| ^{p-2}x')' = bF(x), \] 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.

MSC:
34B40 Boundary value problems on infinite intervals for ordinary differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
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