Even Order Half-Linear Differential Equations with Regularly Varying Coefficients
We establish nonoscillation criterion for the even order half-linear differential equation <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mr...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2020-07-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/8/8/1236 |
| _version_ | 1797561165724778496 |
|---|---|
| author | Vojtěch Růžička |
| author_facet | Vojtěch Růžička |
| author_sort | Vojtěch Růžička |
| collection | DOAJ |
| description | We establish nonoscillation criterion for the even order half-linear differential equation <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msup><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>,</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> are real numbers, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mo>Φ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced open="|" close="|"><mi>s</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>sgn</mi><mi>s</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub></semantics></math></inline-formula> is a regularly varying (at infinity) function of the index <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are replaced by the <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mi>α</mi></msup></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></msup></semantics></math></inline-formula>, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique. |
| first_indexed | 2024-03-10T18:11:20Z |
| format | Article |
| id | doaj.art-a60b21c1784a48e9829b228cc11fc786 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:11:20Z |
| publishDate | 2020-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-a60b21c1784a48e9829b228cc11fc7862023-11-20T08:07:40ZengMDPI AGMathematics2227-73902020-07-0188123610.3390/math8081236Even Order Half-Linear Differential Equations with Regularly Varying CoefficientsVojtěch Růžička0Department of Mathematics and Physics, Faculty of Military Technology, University of Defence in Brno, Kounicova 65, 662 10 Brno, Czech RepublicWe establish nonoscillation criterion for the even order half-linear differential equation <inline-formula><math display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mi>n</mi></msup><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msup><mrow><mo>(</mo><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msup><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><msup><mfenced separators="" open="(" close=")"><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>Φ</mo><mfenced separators="" open="(" close=")"><msup><mi>x</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup></mfenced></mfenced><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mi>l</mi><mo>)</mo></mrow></msup><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math display="inline"><semantics><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>,</mo><msub><mi>β</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>β</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></semantics></math></inline-formula> are real numbers, <inline-formula><math display="inline"><semantics><mrow><mi>n</mi><mo>∈</mo><mi mathvariant="double-struck">N</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mo>Φ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>=</mo><msup><mfenced open="|" close="|"><mi>s</mi></mfenced><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>sgn</mi><mi>s</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math display="inline"><semantics><mrow><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub></semantics></math></inline-formula> is a regularly varying (at infinity) function of the index <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>l</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><mi>α</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. This equation can be understood as a generalization of the even order Euler type half-linear differential equation. We obtain this Euler type equation by rewriting the equation above as follows: the terms <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mi>n</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mi>f</mi><mrow><mi>n</mi><mo>−</mo><mi>l</mi></mrow></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> are replaced by the <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mi>α</mi></msup></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><msup><mi>t</mi><mrow><mi>α</mi><mo>−</mo><mi>l</mi><mi>p</mi></mrow></msup></semantics></math></inline-formula>, respectively. Unlike in other texts dealing with the Euler type equation, in this article an approach based on the theory of regularly varying functions is used. We establish a nonoscillation criterion by utilizing the variational technique.https://www.mdpi.com/2227-7390/8/8/1236higher order half-linear differential equationnonoscillation criterionvariational principleenergy functionalregular variation |
| spellingShingle | Vojtěch Růžička Even Order Half-Linear Differential Equations with Regularly Varying Coefficients Mathematics higher order half-linear differential equation nonoscillation criterion variational principle energy functional regular variation |
| title | Even Order Half-Linear Differential Equations with Regularly Varying Coefficients |
| title_full | Even Order Half-Linear Differential Equations with Regularly Varying Coefficients |
| title_fullStr | Even Order Half-Linear Differential Equations with Regularly Varying Coefficients |
| title_full_unstemmed | Even Order Half-Linear Differential Equations with Regularly Varying Coefficients |
| title_short | Even Order Half-Linear Differential Equations with Regularly Varying Coefficients |
| title_sort | even order half linear differential equations with regularly varying coefficients |
| topic | higher order half-linear differential equation nonoscillation criterion variational principle energy functional regular variation |
| url | https://www.mdpi.com/2227-7390/8/8/1236 |
| work_keys_str_mv | AT vojtechruzicka evenorderhalflineardifferentialequationswithregularlyvaryingcoefficients |