Oscillation and Asymptotic Properties of Differential Equations of Third-Order
The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML&...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-08-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/10/3/192 |
| _version_ | 1797520204828246016 |
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| author | R. Elayaraja V. Ganesan Omar Bazighifan Clemente Cesarano |
| author_facet | R. Elayaraja V. Ganesan Omar Bazighifan Clemente Cesarano |
| author_sort | R. Elayaraja |
| collection | DOAJ |
| description | The main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mi>a</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mrow><mo>[</mo><mo>(</mo><mi>b</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mfenced separators="" open="[" close="]"><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mfenced><mo>′</mo></msup><mo>)</mo></mrow><mo>′</mo></msup><mo>]</mo></mrow><mi>β</mi></msup><msup><mrow><mo>)</mo></mrow><mo>′</mo></msup></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>c</mi></mrow><mi>d</mi></msubsup><mi>q</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><msup><mi>x</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>μ</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>≥</mo><msub><mi>ι</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. New oscillation results are established by using the generalized Riccati technique under the assumption of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>0</mn></msub></mrow><mi>ι</mi></msubsup><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>β</mi></mrow></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mrow></semantics></math></inline-formula><<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>0</mn></msub></mrow><mi>ι</mi></msubsup><mfrac><mn>1</mn><mrow><mi>b</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mfrac><mi>d</mi><mi>s</mi><mo>=</mo><mo>∞</mo><mspace width="4.pt"></mspace><mi>as</mi><mspace width="4.pt"></mspace><mi>ι</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem. |
| first_indexed | 2024-03-10T07:54:28Z |
| format | Article |
| id | doaj.art-b5bb751e8ab14711a9169f40217095fd |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T07:54:28Z |
| publishDate | 2021-08-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-b5bb751e8ab14711a9169f40217095fd2023-11-22T12:02:45ZengMDPI AGAxioms2075-16802021-08-0110319210.3390/axioms10030192Oscillation and Asymptotic Properties of Differential Equations of Third-OrderR. Elayaraja0V. Ganesan1Omar Bazighifan2Clemente Cesarano3Department of Mathematics, Annai Mathammal Sheela Engineering College, Namakkal 637013, Tamil Nadu, IndiaPG and Research Department of Mathematics, Aringar Anna Government Arts College, Namakkal 637002, Tamil Nnadu, IndiaSection of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Rome, ItalySection of Mathematics, International Telematic University Uninettuno, CorsoVittorio Emanuele II, 39, 00186 Rome, ItalyThe main purpose of this study is aimed at developing new criteria of the iterative nature to test the asymptotic and oscillation of nonlinear neutral delay differential equations of third order with noncanonical operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><mi>a</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mrow><mo>[</mo><mo>(</mo><mi>b</mi><mrow><mo>(</mo><mi>ι</mi><mo>)</mo></mrow><msup><mfenced separators="" open="[" close="]"><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mfenced><mo>′</mo></msup><mo>)</mo></mrow><mo>′</mo></msup><mo>]</mo></mrow><mi>β</mi></msup><msup><mrow><mo>)</mo></mrow><mo>′</mo></msup></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>+</mo><msubsup><mo>∫</mo><mrow><mi>c</mi></mrow><mi>d</mi></msubsup><mi>q</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><msup><mi>x</mi><mi>β</mi></msup><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>ι</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>μ</mi><mo>=</mo><mn>0</mn><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ι</mi><mo>≥</mo><msub><mi>ι</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>:</mo><mo>=</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mo>+</mo><mi>p</mi><mo>(</mo><mi>ι</mi><mo>)</mo><mi>x</mi><mo>(</mo><mi>ι</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula>. New oscillation results are established by using the generalized Riccati technique under the assumption of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>0</mn></msub></mrow><mi>ι</mi></msubsup><msup><mi>a</mi><mrow><mo>−</mo><mn>1</mn><mo>/</mo><mi>β</mi></mrow></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>d</mi><mi>s</mi></mrow></semantics></math></inline-formula><<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>0</mn></msub></mrow><mi>ι</mi></msubsup><mfrac><mn>1</mn><mrow><mi>b</mi><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mfrac><mi>d</mi><mi>s</mi><mo>=</mo><mo>∞</mo><mspace width="4.pt"></mspace><mi>as</mi><mspace width="4.pt"></mspace><mi>ι</mi><mo>→</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. Our new results complement the related contributions to the subject. An example is given to prove the significance of new theorem.https://www.mdpi.com/2075-1680/10/3/192oscillationthird-orderneutral differential equationRiccati transformationdistributed deviating arguments |
| spellingShingle | R. Elayaraja V. Ganesan Omar Bazighifan Clemente Cesarano Oscillation and Asymptotic Properties of Differential Equations of Third-Order Axioms oscillation third-order neutral differential equation Riccati transformation distributed deviating arguments |
| title | Oscillation and Asymptotic Properties of Differential Equations of Third-Order |
| title_full | Oscillation and Asymptotic Properties of Differential Equations of Third-Order |
| title_fullStr | Oscillation and Asymptotic Properties of Differential Equations of Third-Order |
| title_full_unstemmed | Oscillation and Asymptotic Properties of Differential Equations of Third-Order |
| title_short | Oscillation and Asymptotic Properties of Differential Equations of Third-Order |
| title_sort | oscillation and asymptotic properties of differential equations of third order |
| topic | oscillation third-order neutral differential equation Riccati transformation distributed deviating arguments |
| url | https://www.mdpi.com/2075-1680/10/3/192 |
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