A Survey on Sharp Oscillation Conditions of Differential Equations with Several Delays

oleh: Mahmoud Abdel-Aty, Musa E. Kavgaci, Ioannis P. Stavroulakis, Nour Zidan

Format: Article
Diterbitkan: MDPI AG 2020-09-01

Deskripsi

This paper deals with the oscillation of the first-order differential equation with several delay arguments <inline-formula><math display="inline"><semantics><mrow><msup><mi>x</mi><mo>′</mo></msup><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>+</mo><msubsup><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>m</mi></msubsup><msub><mi>p</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mi>x</mi><mfenced separators="" open="(" close=")"><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced></mfenced><mo>=</mo><mn>0</mn><mo>,</mo><mspace width="4pt"></mspace><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo></mrow></semantics></math></inline-formula> where the functions <inline-formula><math display="inline"><semantics><mrow><msub><mi>p</mi><mi>i</mi></msub><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mo>∈</mo><mi>C</mi><mfenced separators="" open="(" close=")"><mfenced separators="" open="[" close=")"><msub><mi>t</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo></mfenced><mo>,</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> for every <inline-formula><math display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>m</mi><mo>,</mo><mspace width="3.33333pt"></mspace><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>≤</mo><mi>t</mi></mrow></semantics></math></inline-formula> for <inline-formula><math display="inline"><semantics><mrow><mi>t</mi><mo>≥</mo><msub><mi>t</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mrow><msub><mo movablelimits="true" form="prefix">lim</mo><mrow><mi>t</mi><mo>→</mo><mo>∞</mo></mrow></msub><msub><mi>τ</mi><mi>i</mi></msub><mfenced open="(" close=")"><mi>t</mi></mfenced><mo>=</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. In this paper, the state-of-the-art on the sharp oscillation conditions are presented. In particular, several sufficient oscillation conditions are presented and it is shown that, under additional hypotheses dealing with slowly varying at infinity functions, some of the “liminf” oscillation conditions can be essentially improved replacing “liminf” by “limsup”. The importance of the slowly varying hypothesis and the essential improvement of the sufficient oscillation conditions are illustrated by examples.