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Interval oscillation criteria for second-order forced delay dynamic equations with mixed nonlinearities. (English) Zbl 1197.34117

Summary: Interval oscillation criteria are established for second-order forced delay dynamic equations on time scales containing mixed nonlinearities of the form

${\left(r\left(t\right){{\Phi }}_{\alpha }\left({x}^{{\Delta }}\left(t\right)\right)\right)}^{{\Delta }}+{p}_{0}\left(t\right){{\Phi }}_{\alpha }\left(x\left({\tau }_{0}\left(t\right)\right)\right)+\sum _{i=1}^{n}{p}_{i}\left(t\right){{\Phi }}_{{\beta }_{i}}\left(x\left({\tau }_{i}\left(t\right)\right)\right)=e\left(t\right),\phantom{\rule{1.em}{0ex}}t\in {\left[{t}_{0},\infty \right)}_{𝕋}$

where $𝕋$ is a time scale, ${t}_{0}\in 𝕋$ a fixed number; ${\left[{t}_{0},\infty \right)}_{𝕋}$ is a time scale interval; ${{\Phi }}_{*}\left(u\right)={|u|}^{*-1}u$; the functions $r,{p}_{i},e:{\left[{t}_{0},\infty \right)}_{𝕋}\to ℝ$ are right-dense continuous with $r>0$ nondecreasing; ${\tau }_{k}:𝕋\to 𝕋$ are nondecreasing right-dense continuous with ${\tau }_{k}\left(t\right)\le t$, ${lim}_{t\to \infty }{\tau }_{k}\left(t\right)=\infty$; and the exponents satisfy

${\beta }_{1}\ge \cdots \ge {\beta }_{m}>\alpha >{\beta }_{m+1}\ge \cdots {\beta }_{n}>0·$

All results are new even for $𝕋=ℝ$ and $𝕋=ℤ$. Analogous results for related advance type equations are also given, as well as extended delay and advance equations. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives. Two examples are provided to illustrate one of the theorems.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34N05 Dynamic equations on time scales or measure chains