On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations
This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(&...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/12/1926 |
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| Summary: | This paper mainly investigates a class of third-order semilinear delay differential equations with a nonhomogeneous term <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mo>(</mo><msup><mrow><mo>[</mo><msup><mi>x</mi><mo>″</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>]</mo></mrow><mi>α</mi></msup><mo>)</mo></mrow><mo>′</mo></msup><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msup><mi>x</mi><mi>α</mi></msup><mrow><mo>(</mo><mi>σ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub><mo>.</mo></mrow></semantics></math></inline-formula> Under the oscillation criteria, we propose a sufficient condition to ensure that all solutions for the equation exhibit oscillatory behavior when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> is the quotient of two positive odd integers, supported by concrete examples to verify the accuracy of these conditions. Furthermore, for the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>,</mo></mrow></semantics></math></inline-formula> a sufficient condition is established to guarantee that the solutions either oscillate or asymptotically converge to zero. Moreover, under these criteria, we demonstrate that the global oscillatory behavior of solutions remains unaffected by time-delay functions, nonhomogeneous terms, or nonlinear perturbations when <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi><mo>=</mo><mn>1</mn><mo>.</mo></mrow></semantics></math></inline-formula> Finally, numerical simulations are provided to validate the effectiveness of the derived conclusions. |
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| ISSN: | 2227-7390 |