<abstract xmlns="http://www.w3.org/1999/xhtml">In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0009_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><msup><mrow><mrow><mrow><mo>(</mo><msup><mi>x</mi><mo>′</mo></msup><mo>)</mo></mrow></mrow></mrow><mi>γ</mi></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mi>q</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mi>α</mi></msup><mrow><mo>(</mo><mrow><mi>τ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math><tex-math>{\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0</tex-math></alternatives></disp-formula>Under the assumption <italic>∫∞(r(n))−1/γdη=∞</italic>, we consider the two cases when γ > α and γ &lt; α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.</abstract>

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form
(r(x′)γ)′(t)+q(t)xα(τ(t))=0{\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0Under the assumption ∫∞(r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

Tatra Mountains Mathematical Publications

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

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the...

Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

Published Online: Apr 24, 2020

Page range: 135 - 146

Received: Mar 30, 2019

DOI: https://doi.org/10.2478/tmmp-2020-0009

Keywords
oscillation, non-oscillation, delay, linear, Lebesgue’s dominated convergence theorem

© 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 Oscillation of Second-Order Delay Differential Equations

Published Online: Apr 24, 2020

Page range: 135 - 146

Received: Mar 30, 2019

DOI: https://doi.org/10.2478/tmmp-2020-0009

Keywordsoscillation, non-oscillation, delay, linear, Lebesgue’s dominated convergence theorem

© 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, delay, linear, Lebesgue’s dominated convergence theorem