## Interval oscillation criteria for second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals.(English)Zbl 1226.26020

Summary: By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes integrals of the form
$[p(t)\varphi_\alpha(x^\Delta(t))]^\Delta+ q(t)\varphi_\alpha(x(\tau(t)))+ \int^{\sigma(b)}_a r(t, s)\varphi_{\gamma(s)}(x(g(t, s))) \Delta\xi(s) = e(t),$
where $$t\in [t_0, \infty)_{\mathbb T}= [t_0, \infty)\cap \mathbb T$$, $$\mathbb T$$ is a time scale which is unbounded from above; $$\varphi_*(u)= |u|^*\operatorname{sgn}u$$; $$\gamma: [a,b]_{\mathbb T_1}\rightarrow \mathbb R$$ is a strictly increasing right-dense continuous function; $$p,q,e: [t_0,\infty)_{\mathbb T} \rightarrow\mathbb R$$, $$r:[t_0, \infty)_{\mathbb T}\times [a,b]_{\mathbb T_1}\rightarrow \mathbb R$$, $$\tau:[t_0, \infty)_{\mathbb T}\rightarrow [t_0,\infty)_{\mathbb T}$$, and $$g:[t_0,\infty)_{\mathbb T}\times [a, b]_{\mathbb T_1}\rightarrow [t_0, \infty)_{\mathbb T}$$ are right-dense continuous functions; $$\xi:[a, b]_{\mathbb T_1} \rightarrow \mathbb R$$ is strictly increasing. Some interval oscillation criteria are established in both cases of delayed and advanced arguments. As a special case, the work in this paper unifies and improves many existing results in the literature for equations with a finite number of nonlinear terms.

### MSC:

 2.6e+71 Real analysis on time scales or measure chains

### Keywords:

right-dense continuous function; interval oscillation
Full Text:

### References:

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