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

Sun, Yuangong

Abstr. Appl. Anal. 2011, Article ID 719628, 14 p. (2011).

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:

26E70

Real analysis on time scales or measure chains

Keywords:

right-dense continuous function; interval oscillation

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