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A note on an oscillation criterion for an equation with damped term. (English) Zbl 0542.34028

In this paper a new oscillation criterion is given for the equation (1) \(x''(t)+p(t)x'(t)+q(t)x(t)=0, t\in [t_ 0,\infty)\), where p(t) and q(t) are allowed to change sign on \([t_ 0,\infty)\). The theorem is as follows: Theorem. Suppose for some \(\alpha \in(1,\infty)\) and \(\beta \in [0,1)\), \[ \lim \sup_{t\to \infty}t^{-\alpha}\int^{t}_{t_ 0}(t- s)^{\alpha}s^{\beta}q(s)ds=\infty, \]
\[ \lim \sup_{t\to \infty}t^{-\alpha}\int^{t}_{t_ 0}[(t-s)p(s)s+\alpha s-\beta(t- s)]^ 2(t-s)^{\alpha -2}s^{\beta -2}ds<\infty. \] Then (1) is oscillatory. The above result improves and generalizes Kamenev’s theorem [I. V. Kamenev, Mat. Zametki 23, 249-251 (1978; Zbl 0386.34032)].

MSC:

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

Citations:

Zbl 0386.34032
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References:

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