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

Sun, Yuangong; Kong, Qingkai

Comput. Math. Appl. 62, No. 1, 243-252 (2011).

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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