<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay differential equations of the form
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0008_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mo>[</mo><mrow><mi>r</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>[</mo><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mo>(</mo><mrow><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>p</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>-</mo><mi>τ</mi></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow></mrow><mi>α</mi></msup></mrow><mo>]</mo></mrow><mo>+</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></munderover><mrow><msub><mrow><mi>q</mi></mrow><mi>i</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mi>H</mi><mrow><mo>(</mo><mrow><mi>x</mi><mrow><mo>(</mo><mrow><mi>t</mi><mo>-</mo><msub><mrow><mi>σ</mi></mrow><mi>i</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mi> </mi><mi> </mi><mi> </mi><mtext>for</mtext><mi> </mi><mi>t</mi><mo>≥</mo><msub><mrow><mi>t</mi></mrow><mn>0</mn></msub><mo>></mo><mn>0</mn><mo>,</mo></mrow></mrow></math><tex-math>{d \over {dt}}\left[ {r\left( t \right){{\left[ {{d \over {dt}}\left( {x\left( t \right) + p\left( t \right)x\left( {t - \tau } \right)} \right)} \right]}^\alpha }} \right] + \sum\limits_{i = 1}^m {{q_i}\left( t \right)H\left( {x\left( {t - {\sigma _i}} \right)} \right) = 0\,\,\,{\rm{for}}\,t \ge {t_0} > 0,}</tex-math></alternatives></disp-formula>under the assumption <italic>∫∞(r(n))−1/αdη=∞</italic>. Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.</abstract>

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay differential equations of the form
ddt[r(t)[ddt(x(t)+p(t)x(t-τ))]α]+∑i=1mqi(t)H(x(t-σi))=0   for t≥t0>0,{d \over {dt}}\left[ {r\left( t \right){{\left[ {{d \over {dt}}\left( {x\left( t \right) + p\left( t \right)x\left( {t - \tau } \right)} \right)} \right]}^\alpha }} \right] + \sum\limits_{i = 1}^m {{q_i}\left( t \right)H\left( {x\left( {t - {\sigma _i}} \right)} \right) = 0\,\,\,{\rm{for}}\,t \ge {t_0} > 0,}under the assumption ∫∞(r(n))−1/αdη=∞. Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays

Tatra Mountains Mathematical Publications

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay...

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays

Published Online: Apr 24, 2020

Page range: 121 - 134

Received: Nov 11, 2018

DOI: https://doi.org/10.2478/tmmp-2020-0008

Keywords
oscillation, non-oscillation, neutral, delay, nonlinear, Lebesgue’s dominated convergence theorem, Banach’s contraction principle

© 2020 Shyam Sundar Santra, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays

Published Online: Apr 24, 2020

Page range: 121 - 134

Received: Nov 11, 2018

DOI: https://doi.org/10.2478/tmmp-2020-0008

Keywordsoscillation, non-oscillation, neutral, delay, nonlinear, Lebesgue’s dominated convergence theorem, Banach’s contraction principle

© 2020 Shyam Sundar Santra, published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Keywords
oscillation, non-oscillation, neutral, delay, nonlinear, Lebesgue’s dominated convergence theorem, Banach’s contraction principle