Generalized Prüfer angle and oscillation of half-linear differential equations

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• 作者：O. Do&scaron ; lý ; ^a ; J. Jaro&scaron ; ^b ; M. Veselý ; ^a ; ^{xvesely@math.muni.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Prü ; fer angle ; Half-linear equations ; Oscillation theory ; Conditional oscillation ; Oscillation constant
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：64
• 期：Complete
• 页码：34-41
• 全文大小：769 K

文摘

In this paper, we introduce a new modification of the half-linear Prüfer angle. Applying this modification, we investigate the conditional oscillation of the half-linear second order differential equation where p>1$p>1$, α≠p$\alpha \ne p$, and r,s$r,s$ are continuous functions such that r(t)>0$r\left(t\right)>0$ for large t$t$. We present conditions on the functions r,s$r,s$ which guarantee that Eq. (∗$\ast$) behaves like the Euler type equation [t^α−1Φ(x^′)]^′+λt^α−1−pΦ(x)=0,${\left[t^{\alpha -1}\Phi \left(x^{\prime }\right)\right]}^{\prime }+\lambda t^{\alpha -1-p}\Phi \left(x\right)=0,$ which is conditionally oscillatory with the oscillation constant λ₀=|p−α|^p∕p^p${\lambda }_0=|p-\alpha {|}^p∕p^p$.