Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations
In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup>...
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2024-11-01
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| author | Taher S. Hassan Mnaouer Kachout Bassant M. El-Matary Loredana Florentina Iambor Ismoil Odinaev Akbar Ali |
| author_facet | Taher S. Hassan Mnaouer Kachout Bassant M. El-Matary Loredana Florentina Iambor Ismoil Odinaev Akbar Ali |
| author_sort | Taher S. Hassan |
| collection | DOAJ |
| description | In this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="{" close="}"><msub><mi>α</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><msub><mi>δ</mi><mn>2</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="[" close="]"><msub><mi>α</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>η</mi></mfenced><msub><mi>ϕ</mi><msub><mi>δ</mi><mn>1</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mi>u</mi><mo>Δ</mo></msup><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow></mfenced></mfenced><mo>Δ</mo></msup></mfenced></mfenced><mo>Δ</mo></msup><mo>+</mo><mi>p</mi><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><mi>δ</mi></msub><mfenced separators="" open="(" close=")"><mi>u</mi><mo>(</mo><mi>g</mi><mo>(</mo><mi>η</mi><mo>)</mo><mo>)</mo></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> on an arbitrary unbounded-above time scale <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">T</mi></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>∈</mo><msub><mrow><mo>[</mo><msub><mi>η</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mi mathvariant="double-struck">T</mi></msub><mo>:</mo><mo>=</mo><mrow><mo>[</mo><msub><mi>η</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mo>∩</mo><mi mathvariant="double-struck">T</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>∈</mo><mi mathvariant="double-struck">T</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ϕ</mi><mi>ζ</mi></msub><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mfenced open="|" close="|"><mi>w</mi></mfenced><mi>ζ</mi></msup><mspace width="3.33333pt"></mspace><mi>sgn</mi><mi>w</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Using the integral mean approach and the known Riccati transform methodology, several improved Hille-type and Ohriska-type oscillation criteria have been derived that do not require some restrictive assumptions in the relevant results. Illustrative examples and conclusions show that these criteria are sharp for all third-order dynamic equations compared to the previous results in the literature. |
| format | Article |
| id | doaj-art-e218832aee194b8d9da1393761c3ef36 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-e218832aee194b8d9da1393761c3ef362025-08-20T02:38:47ZengMDPI AGMathematics2227-73902024-11-011223374010.3390/math12233740Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic EquationsTaher S. Hassan0Mnaouer Kachout1Bassant M. El-Matary2Loredana Florentina Iambor3Ismoil Odinaev4Akbar Ali5Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaDepartment of Computer Engineering, College of Computer Science and Engineering, University of Ha’il, Hail 2440, Saudi ArabiaDepartment of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi ArabiaDepartment of Mathematics and Computer Science, University of Oradea, 410087 Oradea, RomaniaDepartment of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, RussiaDepartment of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaIn this paper, we examine the oscillatory behavior of solutions to a class of half-linear third-order dynamic equations with deviating arguments <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="{" close="}"><msub><mi>α</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><msub><mi>δ</mi><mn>2</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="[" close="]"><msub><mi>α</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>η</mi></mfenced><msub><mi>ϕ</mi><msub><mi>δ</mi><mn>1</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mi>u</mi><mo>Δ</mo></msup><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow></mfenced></mfenced><mo>Δ</mo></msup></mfenced></mfenced><mo>Δ</mo></msup><mo>+</mo><mi>p</mi><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><mi>δ</mi></msub><mfenced separators="" open="(" close=")"><mi>u</mi><mo>(</mo><mi>g</mi><mo>(</mo><mi>η</mi><mo>)</mo><mo>)</mo></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> on an arbitrary unbounded-above time scale <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">T</mi></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>η</mi><mo>∈</mo><msub><mrow><mo>[</mo><msub><mi>η</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mi mathvariant="double-struck">T</mi></msub><mo>:</mo><mo>=</mo><mrow><mo>[</mo><msub><mi>η</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mo>∩</mo><mi mathvariant="double-struck">T</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>η</mi><mn>0</mn></msub><mo>∈</mo><mi mathvariant="double-struck">T</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>ϕ</mi><mi>ζ</mi></msub><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mfenced open="|" close="|"><mi>w</mi></mfenced><mi>ζ</mi></msup><mspace width="3.33333pt"></mspace><mi>sgn</mi><mi>w</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. Using the integral mean approach and the known Riccati transform methodology, several improved Hille-type and Ohriska-type oscillation criteria have been derived that do not require some restrictive assumptions in the relevant results. Illustrative examples and conclusions show that these criteria are sharp for all third-order dynamic equations compared to the previous results in the literature.https://www.mdpi.com/2227-7390/12/23/3740oscillation criteriaHille-typeOhriska-typedifferential equationsdynamic equationstime scales |
| spellingShingle | Taher S. Hassan Mnaouer Kachout Bassant M. El-Matary Loredana Florentina Iambor Ismoil Odinaev Akbar Ali Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations Mathematics oscillation criteria Hille-type Ohriska-type differential equations dynamic equations time scales |
| title | Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations |
| title_full | Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations |
| title_fullStr | Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations |
| title_full_unstemmed | Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations |
| title_short | Improved Hille-Type and Ohriska-Type Criteria for Half-Linear Third-Order Dynamic Equations |
| title_sort | improved hille type and ohriska type criteria for half linear third order dynamic equations |
| topic | oscillation criteria Hille-type Ohriska-type differential equations dynamic equations time scales |
| url | https://www.mdpi.com/2227-7390/12/23/3740 |
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