Oscillation criteria for second-order forced dynamic equations with mixed nonlinearities. (English) Zbl 1181.34099

Agarwal, Ravi P.; Zafer, A.

Adv. Difference Equ. 2009, Article ID 938706, 20 p. (2009).

Summary: We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form
\[ (r(t)\Phi_\alpha(x^\Delta))^\Delta+f(t,x^\sigma)=e(t),\quad t\in[t_0,\infty)_{\mathbb T} \] with
\[ f(t,x)=q(t)\Phi_\alpha(x)+\sum_{i=1}^n q_i(t)\Phi_{\beta_i}(x),\;\Phi_*(u)=|u|^{*-1}u, \]
where \([t_0,\infty)_{\mathbb T}\) is a time scale interval with \(t_0\in \mathbb T\), the functions \(r,q,q_i,e:[t_0,\infty)_{\mathbb T} \to\mathbb R\) are right-dense continuous with \(r>0\), \(\sigma\) is the forward jump operator, \(x^\sigma(t):=x(\sigma(t))\), and \(\beta_1>\cdots>\beta_m>\alpha>\beta_{m+1}>\cdots \beta_n>0\). All results obtained are new even for \(\mathbb T=\mathbb R\); and \(\mathbb T=\mathbb Z\).

MSC:

34N05	Dynamic equations on time scales or measure chains
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

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