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Oscillation criteria of even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals. (English) Zbl 07022304
Summary: We study the oscillatory properties of the following even order delay dynamic equations with nonlinearities given by Riemann-Stieltjes integrals: $$(p(t) \left|x^{\Delta^{n - 1}}(t)\right|^{\alpha - 1} x^{\Delta^{n - 1}}(t))^{\Delta} + f(t, x(\delta(t))) + \int_a^{\sigma(b)} k(t, s) \left|x(g(t, s))\right|^{\theta(s)} \operatorname{sn}(x(g(t, s))) \Delta \xi(s) = 0$$, where $$t \in [t_0, \infty)_{\mathbb{T}} : = [t_0, \infty) \cap \mathbb{T}$$, $$\mathbb{T}$$ a time scale which is unbounded above, $$n \geqslant 2$$ is even, $$\left|f(t, u) \left|\geqslant q(t)\right| u^\alpha\right|$$, $$\alpha > 0$$ is a constant, and $$\theta : [a, b]_{\mathbb{T}_1} \rightarrow \mathbb{R}$$ is a strictly increasing right-dense continuous function; $$p, q : [t_0, \infty)_{\mathbb{T}} \rightarrow \mathbb{R}$$, $$k : [t_0, \infty)_{\mathbb{T}} \times [a, b]_{\mathbb{T}_1} \rightarrow \mathbb{R}$$, $$\delta : [t_0, \infty)_{\mathbb{T}} \rightarrow [t_0, \infty)_{\mathbb{T}}$$, 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. Our results extend and supplement some known results in the literature.

##### MSC:
 34 Ordinary differential equations 39 Difference and functional equations
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