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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2025-06-01
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| author | Wenjin Li Jiaxuan Sun Yanni Pang |
| author_facet | Wenjin Li Jiaxuan Sun Yanni Pang |
| author_sort | Wenjin Li |
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| description | 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. |
| format | Article |
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| institution | Matheson Library |
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| spelling | doaj-art-ab3a5a14bd1b43fead7ee931d90ff7942025-06-25T14:08:39ZengMDPI AGMathematics2227-73902025-06-011312192610.3390/math13121926On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential EquationsWenjin Li0Jiaxuan Sun1Yanni Pang2School of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, ChinaSchool of Applied Mathematics, Jilin University of Finance and Economics, Changchun 130117, ChinaSchool of Mathematics, Jilin University, Changchun 130021, ChinaThis 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.https://www.mdpi.com/2227-7390/13/12/1926oscillation criterianonhomogeneousdelay differential equations |
| spellingShingle | Wenjin Li Jiaxuan Sun Yanni Pang On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations Mathematics oscillation criteria nonhomogeneous delay differential equations |
| title | On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations |
| title_full | On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations |
| title_fullStr | On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations |
| title_full_unstemmed | On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations |
| title_short | On the Study of Solutions for a Class of Third-Order Semilinear Nonhomogeneous Delay Differential Equations |
| title_sort | on the study of solutions for a class of third order semilinear nonhomogeneous delay differential equations |
| topic | oscillation criteria nonhomogeneous delay differential equations |
| url | https://www.mdpi.com/2227-7390/13/12/1926 |
| work_keys_str_mv | AT wenjinli onthestudyofsolutionsforaclassofthirdordersemilinearnonhomogeneousdelaydifferentialequations AT jiaxuansun onthestudyofsolutionsforaclassofthirdordersemilinearnonhomogeneousdelaydifferentialequations AT yannipang onthestudyofsolutionsforaclassofthirdordersemilinearnonhomogeneousdelaydifferentialequations |