A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays
This paper deals with the oscillation of the first-order differential equation with several delay arguments <inline-formula><math display="inline"><semantics><mrow><msup><mi>x</mi><mo>′</mo></msup><mfenced open="(" cl...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/9/1492 |
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| author | Mahmoud Abdel-Aty Musa E. Kavgaci Ioannis P. Stavroulakis Nour Zidan |
| author_facet | Mahmoud Abdel-Aty Musa E. Kavgaci Ioannis P. Stavroulakis Nour Zidan |
| author_sort | Mahmoud Abdel-Aty |
| collection | DOAJ |
| description | This paper deals with the oscillation of the first-order differential equation with several delay arguments <inline-formula><math display="inline"><semantics><mrow><msup><mi>x</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mi>x</mi><mfenced separators="" open="(" close=")"><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> where the functions <inline-formula><math display="inline"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mo>∈</mo><mi>C</mi><mfenced separators="" open="(" close=")"><mfenced separators="" open="[" close=")"><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo></mfenced><mo>,</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> for every <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>≤</mo><mi>t</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>t</mi><mo>→</mo><mo>∞</mo></mrow></msub><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples. |
| first_indexed | 2024-03-10T16:35:50Z |
| format | Article |
| id | doaj.art-6b023eb68964402d93026c3f3a2227ae |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T16:35:50Z |
| publishDate | 2020-09-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-6b023eb68964402d93026c3f3a2227ae2023-11-20T12:27:38ZengMDPI AGMathematics2227-73902020-09-0189149210.3390/math8091492A Survey on Sharp Oscillation Conditions of Differential Equations with Several DelaysMahmoud Abdel-Aty0Musa E. Kavgaci1Ioannis P. Stavroulakis2Nour Zidan3Department of Mathematics, Faculty of Science, Sohag University, Sohag 82749, EgyptDepartment of Mathematics, Faculty of Science, Ankara University, Tandogan Ankara 06100, TurkeyDepartment of Mathematics, Faculty of Science, University of Ioannina, 451 10 Ioannina, GreeceDepartment of Mathematics, College of Science, Jouf University, Sakaka 42421, Saudi ArabiaThis paper deals with the oscillation of the first-order differential equation with several delay arguments <inline-formula><math display="inline"><semantics><mrow><msup><mi>x</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mi>x</mi><mfenced separators="" open="(" close=")"><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> where the functions <inline-formula><math display="inline"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mo>∈</mo><mi>C</mi><mfenced separators="" open="(" close=")"><mfenced separators="" open="[" close=")"><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo></mfenced><mo>,</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> for every <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>≤</mo><mi>t</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>t</mi><mo>→</mo><mo>∞</mo></mrow></msub><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.https://www.mdpi.com/2227-7390/8/9/1492oscillationdelay argumentsdifferential equations |
| spellingShingle | Mahmoud Abdel-Aty Musa E. Kavgaci Ioannis P. Stavroulakis Nour Zidan A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays Mathematics oscillation delay arguments differential equations |
| title | A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays |
| title_full | A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays |
| title_fullStr | A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays |
| title_full_unstemmed | A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays |
| title_short | A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays |
| title_sort | survey on sharp oscillation conditions of differential equations with several delays |
| topic | oscillation delay arguments differential equations |
| url | https://www.mdpi.com/2227-7390/8/9/1492 |
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