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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| Main Authors: | , , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2024-11-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/12/23/3740 |
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| Summary: | 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. |
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| ISSN: | 2227-7390 |