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Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities. (English) Zbl 1125.34024

The authors introduce general oscillation criteria for the second order ordinary differential equation \[ \big(p(t)\,x'\big)'+q(t)\,x+\sum_{i=1}^nq_i(t)\,| x| ^{\alpha_1}\,\mathrm{sgn}\,x=e(t), \] where \(p,q,q_i,e\in C[0,\infty)\), \(p(t)>0\) and differentiable (but this assumption on the existence of \(p'(t)\) is apparently not needed), \(\alpha_1>\dots>\alpha_m>1>\alpha_{m+1}>\dots>\alpha_n\), and no restriction is invoked on the forcing term \(e(t)\). Note that the equation contains both sublinear and superlinear terms due to the assumptions on the exponents \(\alpha_i\). The main results (Theorems 1–3) are derived via the Riccati technique and generalize several results in the literature.

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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