Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order
This paper aims to improve Hille oscillation criteria for the third-order functional dynamic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open=&q...
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MDPI AG
2022-03-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/7/1078 |
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| author | Taher S. Hassan Rabie A. Ramadan Zainab Alsheekhhussain Ahmed Y. Khedr Amir Abdel Menaem Ismoil Odinaev |
| author_facet | Taher S. Hassan Rabie A. Ramadan Zainab Alsheekhhussain Ahmed Y. Khedr Amir Abdel Menaem Ismoil Odinaev |
| author_sort | Taher S. Hassan |
| collection | DOAJ |
| description | This paper aims to improve Hille oscillation criteria for the third-order functional dynamic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="{" close="}"><msub><mi>p</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><msub><mi>γ</mi><mn>2</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="[" close="]"><msub><mi>p</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>ξ</mi></mfenced><msub><mi>ϕ</mi><msub><mi>γ</mi><mn>1</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mi>y</mi><mo>Δ</mo></msup><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mfenced></mfenced><mo>Δ</mo></msup></mfenced></mfenced><mo>Δ</mo></msup><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><mi>γ</mi></msub><mfenced separators="" open="(" close=")"><mi>y</mi><mo>(</mo><mi>g</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>)</mo></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> on an above-unbounded 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>. The obtained results improve related contributions reported in the literature without restrictive conditions on the time scales. To demonstrate the essential results, an example is presented. |
| first_indexed | 2024-03-09T11:38:55Z |
| format | Article |
| id | doaj.art-23a926eabc904f25a495512c9298072d |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:38:55Z |
| publishDate | 2022-03-01 |
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| spelling | doaj.art-23a926eabc904f25a495512c9298072d2023-11-30T23:36:50ZengMDPI AGMathematics2227-73902022-03-01107107810.3390/math10071078Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-OrderTaher S. Hassan0Rabie A. Ramadan1Zainab Alsheekhhussain2Ahmed Y. Khedr3Amir Abdel Menaem4Ismoil Odinaev5Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaCollege of Computer Science and Engineering, University of Ha’il, Ha’il 81481, Saudi ArabiaDepartment of Mathematics, Faculty of Science, University of Ha’il, Ha’il 2440, Saudi ArabiaCollege of Computer Science and Engineering, University of Ha’il, Ha’il 81481, Saudi ArabiaElectrical Engineering Department, Mansoura University, Mansoura 35516, EgyptDepartment of Automated Electrical Systems, Ural Power Engineering Institute, Ural Federal University, 620002 Yekaterinburg, RussiaThis paper aims to improve Hille oscillation criteria for the third-order functional dynamic equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="{" close="}"><msub><mi>p</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><msub><mi>γ</mi><mn>2</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mfenced separators="" open="[" close="]"><msub><mi>p</mi><mn>1</mn></msub><mfenced open="(" close=")"><mi>ξ</mi></mfenced><msub><mi>ϕ</mi><msub><mi>γ</mi><mn>1</mn></msub></msub><mfenced separators="" open="(" close=")"><msup><mi>y</mi><mo>Δ</mo></msup><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow></mfenced></mfenced><mo>Δ</mo></msup></mfenced></mfenced><mo>Δ</mo></msup><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>ξ</mi><mo>)</mo></mrow><msub><mi>ϕ</mi><mi>γ</mi></msub><mfenced separators="" open="(" close=")"><mi>y</mi><mo>(</mo><mi>g</mi><mo>(</mo><mi>ξ</mi><mo>)</mo><mo>)</mo></mfenced><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> on an above-unbounded 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>. The obtained results improve related contributions reported in the literature without restrictive conditions on the time scales. To demonstrate the essential results, an example is presented.https://www.mdpi.com/2227-7390/10/7/1078oscillation criteriathird orderdynamic equationstime scales |
| spellingShingle | Taher S. Hassan Rabie A. Ramadan Zainab Alsheekhhussain Ahmed Y. Khedr Amir Abdel Menaem Ismoil Odinaev Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order Mathematics oscillation criteria third order dynamic equations time scales |
| title | Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order |
| title_full | Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order |
| title_fullStr | Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order |
| title_full_unstemmed | Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order |
| title_short | Improved Hille Oscillation Criteria for Nonlinear Functional Dynamic Equations of Third-Order |
| title_sort | improved hille oscillation criteria for nonlinear functional dynamic equations of third order |
| topic | oscillation criteria third order dynamic equations time scales |
| url | https://www.mdpi.com/2227-7390/10/7/1078 |
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