Neutral Differential Equations of Second-Order: Iterative Monotonic Properties
In this work, we investigate the oscillatory properties of the neutral differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo stretchy="false">(&l...
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MDPI AG
2022-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/9/1356 |
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| author | Osama Moaaz Fahd Masood Clemente Cesarano Shami A. M. Alsallami E. M. Khalil Mohamed L. Bouazizi |
| author_facet | Osama Moaaz Fahd Masood Clemente Cesarano Shami A. M. Alsallami E. M. Khalil Mohamed L. Bouazizi |
| author_sort | Osama Moaaz |
| collection | DOAJ |
| description | In this work, we investigate the oscillatory properties of the neutral differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo stretchy="false">(</mo><mi>r</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo>[</mo><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mi>p</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>g</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>′</mo></msup><mo>]</mo></mrow><mi mathvariant="fraktur">v</mi></msup><mo stretchy="false">)</mo></mrow><mo>′</mo></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>q</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mi>s</mi><mi mathvariant="fraktur">v</mi></msup><mrow><mo stretchy="false">(</mo><msub><mi>h</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>≥</mo><msub><mi>s</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation. |
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| format | Article |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:57:59Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-dbf9b0eaaf0349318d8a45c7de20038f2023-11-23T08:43:04ZengMDPI AGMathematics2227-73902022-04-01109135610.3390/math10091356Neutral Differential Equations of Second-Order: Iterative Monotonic PropertiesOsama Moaaz0Fahd Masood1Clemente Cesarano2Shami A. M. Alsallami3E. M. Khalil4Mohamed L. Bouazizi5Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraydah 51452, Saudi ArabiaDepartment of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, EgyptSection of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Roma, ItalyDepartment of Mathematical Sciences, College of Applied Science, Umm Al-Qura University, Makkah 21955, Saudi ArabiaDepartment of Mathematics and Statistics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi ArabiaDepartment of Mechanical Engineering, College of Engineering, Prince Sattam bin Abdulaziz University, Alkharj 16273, Saudi ArabiaIn this work, we investigate the oscillatory properties of the neutral differential equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo stretchy="false">(</mo><mi>r</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo>[</mo><msup><mrow><mo stretchy="false">(</mo><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><mi>p</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mi>s</mi><mrow><mo stretchy="false">(</mo><mi>g</mi><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>′</mo></msup><mo>]</mo></mrow><mi mathvariant="fraktur">v</mi></msup><mo stretchy="false">)</mo></mrow><mo>′</mo></msup><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></msubsup><msub><mi>q</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><msup><mi>s</mi><mi mathvariant="fraktur">v</mi></msup><mrow><mo stretchy="false">(</mo><msub><mi>h</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>≥</mo><msub><mi>s</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>. We first present new monotonic properties for the solutions of this equation, and these properties are characterized by an iterative nature. Using these new properties, we obtain new oscillation conditions that guarantee that all solutions are oscillate. Our results are a complement and extension to the relevant results in the literature. We test the significance of the results by applying them to special cases of the studied equation.https://www.mdpi.com/2227-7390/10/9/1356Emden–Fowlerneutral differential equationsoscillationnon-canonical |
| spellingShingle | Osama Moaaz Fahd Masood Clemente Cesarano Shami A. M. Alsallami E. M. Khalil Mohamed L. Bouazizi Neutral Differential Equations of Second-Order: Iterative Monotonic Properties Mathematics Emden–Fowler neutral differential equations oscillation non-canonical |
| title | Neutral Differential Equations of Second-Order: Iterative Monotonic Properties |
| title_full | Neutral Differential Equations of Second-Order: Iterative Monotonic Properties |
| title_fullStr | Neutral Differential Equations of Second-Order: Iterative Monotonic Properties |
| title_full_unstemmed | Neutral Differential Equations of Second-Order: Iterative Monotonic Properties |
| title_short | Neutral Differential Equations of Second-Order: Iterative Monotonic Properties |
| title_sort | neutral differential equations of second order iterative monotonic properties |
| topic | Emden–Fowler neutral differential equations oscillation non-canonical |
| url | https://www.mdpi.com/2227-7390/10/9/1356 |
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