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On the oscillation of certain second-order differential equations. (English) Zbl 0958.34027
The authors establish new criteria for the oscillation of all solutions to second-order equations \[ (a(t)|x'(t)|^{\sigma-1}x'(t))+q(t)|x(t)|^\sigma \text{ sgn} x(t)=0,\quad \sigma>1, \] where the functions \(a,q:[t_0,\infty)\to{\mathbb{R}}\) are continuous and \(a(t)>0\) for \(t\geq t_0\). Here, new oscillation criteria are also established for the more general equation \[ (a(t)|x'(t)|^{\sigma-1}x'(t))+p(t)|x'(t)|x'(t)+q(t)f(x(t))=0,\quad \sigma>1, \] where \(a,p,q:[t_0,\infty)\to{\mathbb{R}}\), \(f:{\mathbb{R}}\to{\mathbb{R}}\) are continuous, \(a(t)>0\) and \(xf(x)>0\) for \(u\neq 0\).

MSC:
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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