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**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 |

### Keywords:

interval criteria; forced oscillation; Riemann-Stieltjes integral; nonlinear differential equations
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\textit{Y. Sun} and \textit{Q. Kong}, Comput. Math. Appl. 62, No. 1, 243--252 (2011; Zbl 1228.34055)

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### References:

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