<abstract xmlns="http://www.w3.org/1999/xhtml">In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0025_eq_001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>z</mi><mo>′</mo></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>γ</mi></msup></mrow><mo>)</mo></mrow></mrow></mrow><mo>′</mo></msup><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>σ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><mi>t</mi><mo>≥</mo><msub><mrow><mi>t</mi></mrow><mn>0</mn></msub><mo>,</mo><mi> </mi><mi> </mi><mi> </mi><mi>t</mi><mo>≠</mo><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub><mo>,</mo></mrow></mtd></mtr><mtr><mtd><mrow><mi>Δ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mrow><mrow><mo>(</mo><mrow><msup><mi>z</mi><mo>′</mo></msup><mrow><mo>(</mo><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mi>γ</mi></msup></mrow><mo>)</mo></mrow><mo>+</mo><mi>h</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mi>α</mi></msup><mrow><mo>(</mo><mrow><mi>σ</mi><mrow><mo>(</mo><mrow><msub><mrow><mi>λ</mi></mrow><mi>k</mi></msub></mrow><mo>)</mo></mrow></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo></mrow></mtd><mtd><mrow><mi>k</mi><mo>∈</mo><mi>𝕅</mi></mrow></mtd></mtr></mtable></mrow></mrow></mrow></math><tex-math>\left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill &amp; {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill &amp; {k \in \mathbb{N}} \hfill \cr } } \right.</tex-math></alternatives></disp-formula> are established, where
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0025_eq_002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mi>z</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><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>τ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math><tex-math>z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right)</tex-math></alternatives></disp-formula>Under the assumption <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_tmmp-2020-0025_eq_003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mrow><mo>∫</mo><mrow><msup><mrow></mrow><mo>∞</mo></msup><msup><mrow><mrow><mrow><mo>(</mo><mrow><mi>r</mi><mrow><mo>(</mo><mi>η</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></mrow><mrow><mo>-</mo><mn>1</mn><mo>/</mo><mi>α</mi></mrow></msup><mi>d</mi><mi>η</mi><mo>=</mo><mo>∞</mo></mrow></mrow></mrow></math><tex-math>\int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty }</tex-math></alternatives></inline-formula> two cases when <italic>γ>α</italic> and <italic>γ&lt;α</italic> are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.</abstract>

In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential system{(r(t)(z′(t))γ)′+q(t)xα(σ(t))=0,t≥t0,   t≠λk,Δ(r(λk)(z′(λk))γ)+h(λk)xα(σ(λk))=0,k∈𝕅\left\{ {\matrix{{{{\left( {r\left( t \right){{\left( {z'\left( t \right)} \right)}^\gamma }} \right)}^\prime } + q\left( t \right){x^\alpha }\left( {\sigma \left( t \right)} \right) = 0,} \hfill & {t \ge {t_0},\,\,\,t \ne {\lambda _k},} \hfill \cr {\Delta \left( {r\left( {{\lambda _k}} \right){{\left( {z'\left( {{\lambda _k}} \right)} \right)}^\gamma }} \right) + h\left( {{\lambda _k}} \right){x^\alpha }\left( {\sigma \left( {{\lambda _k}} \right)} \right) = 0,} \hfill & {k \in \mathbb{N}} \hfill \cr } } \right. are established, where
z(t)=x(t)+p(t)x(τ(t))z\left( t \right) = x\left( t \right) + p\left( t \right)x\left( {\tau \left( t \right)} \right)Under the assumption ∫∞(r(η))-1/αdη=∞\int {^\infty {{\left( {r\left( \eta \right)} \right)}^{ - 1/\alpha }}d\eta = \infty } two cases when γ>α and γ<α are considered. The main tool is Lebesgue’s Dominated Convergence theorem. Examples are given to illustrate the main results, and state an open problem.

Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

Tatra Mountains Mathematical Publications

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

In this work, necessary and sufficient conditions for oscillation of solutions of second-order neutral impulsive differential...

Necessary and Sufficient Conditions for Oscillation of Solutions to Second-Order Neutral Differential Equations with Impulses

Published Online: Nov 04, 2020

Page range: 157 - 170

Received: May 18, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0025

Keywords
Oscillation, non-oscillation, neutral, delay, Lebesgue’s Dominated Convergence theorem, impulses

© 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 Solutions to Second-Order Neutral Differential Equations with Impulses

Published Online: Nov 04, 2020

Page range: 157 - 170

Received: May 18, 2020

DOI: https://doi.org/10.2478/tmmp-2020-0025

KeywordsOscillation, non-oscillation, neutral, delay, Lebesgue’s Dominated Convergence theorem, impulses

© 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, Lebesgue’s Dominated Convergence theorem, impulses