Hille-Kneser-type criteria for second-order dynamic equations on time scales
<p/> <p>We consider the pair of second-order dynamic equations, (<it>r</it>(<it>t</it>)(<it>x</it><sup>Δ</sup>)<sup><it>γ</it></sup>)<sup>Δ</sup> + <it>p</it>(<it...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2006-01-01
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| Series: | Advances in Difference Equations |
| Online Access: | http://www.advancesindifferenceequations.com/content/2006/051401 |
| _version_ | 1811264174195474432 |
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| author | Saker SH Erbe L Peterson A |
| author_facet | Saker SH Erbe L Peterson A |
| author_sort | Saker SH |
| collection | DOAJ |
| description | <p/> <p>We consider the pair of second-order dynamic equations, (<it>r</it>(<it>t</it>)(<it>x</it><sup>Δ</sup>)<sup><it>γ</it></sup>)<sup>Δ</sup> + <it>p</it>(<it>t</it>)<it>x</it><sup><it>γ</it></sup>(<it>t</it>) = 0 and (<it>r</it>(<it>t</it>)(<it>x</it><sup>Δ</sup>)<sup><it>γ</it></sup>)<sup>Δ</sup> + <it>p</it>(<it>t</it>)<it>x</it><sup><it>γσ</it></sup>(<it>t</it>) = 0, on a time scale <inline-formula><graphic file="1687-1847-2006-051401-i1.gif"/></inline-formula>, where <it>γ</it> > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when <inline-formula><graphic file="1687-1847-2006-051401-i2.gif"/></inline-formula> involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results.</p> |
| first_indexed | 2024-04-12T19:58:24Z |
| format | Article |
| id | doaj.art-f3cf926c34384fc2b7938787829b7832 |
| institution | Directory Open Access Journal |
| issn | 1687-1839 1687-1847 |
| language | English |
| last_indexed | 2024-04-12T19:58:24Z |
| publishDate | 2006-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-f3cf926c34384fc2b7938787829b78322022-12-22T03:18:36ZengSpringerOpenAdvances in Difference Equations1687-18391687-18472006-01-0120061051401Hille-Kneser-type criteria for second-order dynamic equations on time scalesSaker SHErbe LPeterson A<p/> <p>We consider the pair of second-order dynamic equations, (<it>r</it>(<it>t</it>)(<it>x</it><sup>Δ</sup>)<sup><it>γ</it></sup>)<sup>Δ</sup> + <it>p</it>(<it>t</it>)<it>x</it><sup><it>γ</it></sup>(<it>t</it>) = 0 and (<it>r</it>(<it>t</it>)(<it>x</it><sup>Δ</sup>)<sup><it>γ</it></sup>)<sup>Δ</sup> + <it>p</it>(<it>t</it>)<it>x</it><sup><it>γσ</it></sup>(<it>t</it>) = 0, on a time scale <inline-formula><graphic file="1687-1847-2006-051401-i1.gif"/></inline-formula>, where <it>γ</it> > 0 is a quotient of odd positive integers. We establish some necessary and sufficient conditions for nonoscillation of Hille-Kneser type. Our results in the special case when <inline-formula><graphic file="1687-1847-2006-051401-i2.gif"/></inline-formula> involve the well-known Hille-Kneser-type criteria of second-order linear differential equations established by Hille. For the case of the second-order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time scales. Several examples are considered to illustrate the main results.</p>http://www.advancesindifferenceequations.com/content/2006/051401 |
| spellingShingle | Saker SH Erbe L Peterson A Hille-Kneser-type criteria for second-order dynamic equations on time scales Advances in Difference Equations |
| title | Hille-Kneser-type criteria for second-order dynamic equations on time scales |
| title_full | Hille-Kneser-type criteria for second-order dynamic equations on time scales |
| title_fullStr | Hille-Kneser-type criteria for second-order dynamic equations on time scales |
| title_full_unstemmed | Hille-Kneser-type criteria for second-order dynamic equations on time scales |
| title_short | Hille-Kneser-type criteria for second-order dynamic equations on time scales |
| title_sort | hille kneser type criteria for second order dynamic equations on time scales |
| url | http://www.advancesindifferenceequations.com/content/2006/051401 |
| work_keys_str_mv | AT sakersh hilleknesertypecriteriaforsecondorderdynamicequationsontimescales AT erbel hilleknesertypecriteriaforsecondorderdynamicequationsontimescales AT petersona hilleknesertypecriteriaforsecondorderdynamicequationsontimescales |