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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
\[ (r(t)\Phi_\alpha(x^\Delta(t)))^\Delta+p_0(t)\Phi_\alpha(x(\tau_0(t)))+\sum^n_{i=1}p_i(t)\Phi_{\beta_i}(x(\tau_i(t)))=e(t),\quad t\in[t_0,\infty)_{\mathbb T} \]
where \(\mathbb T\) is a time scale, \(t_0\in\mathbb T\) a fixed number; \([t_0,\infty)_{\mathbb T}\) is a time scale interval; \(\Phi_*(u)=|u|^{*-1}u\); the functions \(r,p_i,e:[t_0,\infty)_{\mathbb T}\to\mathbb R\) are right-dense continuous with \(r>0\) nondecreasing; \(\tau_k:\mathbb T\to\mathbb T\) are nondecreasing right-dense continuous with \(\tau_k(t)\leq t\), \(\lim_{t\to\infty}\tau_k(t)=\infty\); and the exponents satisfy
\[ \beta_1\geq\cdots\geq \beta_m>\alpha>\beta_{m+1}\geq \cdots\beta_n>0. \]
All results are new even for \(\mathbb T=\mathbb R\) and \(\mathbb T=\mathbb Z\). 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
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