Symmetry and Its Role in Oscillation of Solutions of Third-Order Differential Equations
Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form <inline-formula><mat...
| Main Authors: | , , , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2021-08-01
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| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/13/8/1485 |
| Summary: | Symmetry plays an essential role in determining the correct methods for the oscillatory properties of solutions to differential equations. This paper examines some new oscillation criteria for unbounded solutions of third-order neutral differential equations of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo stretchy="false">(</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>r</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow><msup><mrow><mo stretchy="false">(</mo><msup><mi>z</mi><mo>′</mo></msup><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><msub><mi>β</mi><mn>1</mn></msub></msup><mo stretchy="false">)</mo></mrow><mo>′</mo></msup><mo stretchy="false">)</mo></mrow><msub><mi>β</mi><mn>2</mn></msub></msup><mo stretchy="false">)</mo></mrow><mo>′</mo></msup></semantics></math></inline-formula> + <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover></mstyle><msub><mi>q</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow><msup><mi>x</mi><msub><mi>β</mi><mn>3</mn></msub></msup><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi><mi>i</mi></msub><mrow><mo stretchy="false">(</mo><mi>ς</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>0</mn><mo>.</mo></mrow></semantics></math></inline-formula> New oscillation results are established by using generalized Riccati substitution, an integral average technique in the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems. |
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| ISSN: | 2073-8994 |