Generalized Prüfer angle and oscillation of half-linear differential equations

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• 作者：O. Do&scaron ; l&yacute ; ^a ; J. Jaro&scaron ; ^b ; M. Vesel&yacute ; ^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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si3.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=9f59afebb137cf8d4a05c9c510a323ef" title="Click to view the MathML source">p>1class="mathContainer hidden">class="mathCode">$p>1$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si4.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=ff04176d35d24d6d4ca50363fa0918a0" title="Click to view the MathML source">α≠pclass="mathContainer hidden">class="mathCode">$\alpha \ne p$, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si5.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=c21b06f2ab5159b853cdc3e971afc699" title="Click to view the MathML source">r,sclass="mathContainer hidden">class="mathCode">$r,s$ are continuous functions such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si6.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=48c17955a5fe10966ffaedc903907e76" title="Click to view the MathML source">r(t)>0class="mathContainer hidden">class="mathCode">$r\left(t\right)>0$ for large class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si7.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=e56cbccb8877b8dd82aadb116faf067d" title="Click to view the MathML source">tclass="mathContainer hidden">class="mathCode">$t$. We present conditions on the functions class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si5.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=c21b06f2ab5159b853cdc3e971afc699" title="Click to view the MathML source">r,sclass="mathContainer hidden">class="mathCode">$r,s$ which guarantee that Eq. (class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si1.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=d1143a2d5ddc29e0f7a576d58bff7d69" title="Click to view the MathML source">∗class="mathContainer hidden">class="mathCode">$\ast$) behaves like the Euler type equation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si10.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=37d5ffa341ec6e87f0c3420514d940e4" title="Click to view the MathML source">[t^α−1Φ(x^′)]^′+λt^α−1−pΦ(x)=0,class="mathContainer hidden">class="mathCode">${\left[t^{\alpha -1}\Phi \left(x^{\prime }\right)\right]}^{\prime }+\lambda t^{\alpha -1-p}\Phi \left(x\right)=0,$ which is conditionally oscillatory with the oscillation constant class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302397&_mathId=si11.gif&_user=111111111&_pii=S0893965916302397&_rdoc=1&_issn=08939659&md5=0a38631064a22f611f138bcef5875a8f" title="Click to view the MathML source">λ₀=|p−α|^p∕p^pclass="mathContainer hidden">class="mathCode">${\lambda }_0=|p-\alpha {|}^p∕p^p$.