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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