## Interval criteria for forced oscillation with nonlinearities given by Riemann-Stieltjes integrals.(English)Zbl 1228.34055

Summary: We study the oscillation of second-order forced differential equations with nonlinearity given by a Riemann-Stieltjes integral of the form $(p(t)x')'+q(t)x+\int_0^br(t,s)|x(t)|^{\alpha(s)}\mathrm{sgn} x(t)d\xi(s)=e(t),$ where $$b\in (0,\infty ]$$, $$\alpha \in C[0,b)$$ is strictly increasing such that $$0\leq \alpha (0)<1<\alpha (b - )$$, $$p,q,e\in C[0,\infty )$$ with $$p(t)>0$$, $$r\in C([0,\infty )\times [0,b))$$, and $$\xi :[0,b)\to \mathbb{R}$$ is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. As a special case, the work in this paper unifies and improves the existing results in the literature for equations with a finite number of nonlinear terms. We also extend our results to equations with delays.

### MSC:

 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34K11 Oscillation theory of functional-differential equations 26A42 Integrals of Riemann, Stieltjes and Lebesgue type
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### References:

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