Oscillation Criteria of Second-Order Dynamic Equations on Time Scales

• Main Authors: Ya-Ru Zhu, Zhong-Xuan Mao, Shi-Pu Liu, Jing-Feng Tian
• Format: Article
• Language: English
• Published: MDPI AG 2021-08-01
• Series: Mathematics
• Subjects: second order dynamic equation; oscillation; nonlinear equation; Riccati technique; delta derivative
• Online Access: https://www.mdpi.com/2227-7390/9/16/1867

In this paper, we consider the oscillation behavior of the following second-order nonlinear dynamic equation. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><mi>λ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi mathvariant="sans-serif">Ψ</mi><mfenced separators="" open="(" close=")"><mfrac><mn>1</mn><mrow><msup><mi>φ</mi><mo>Δ</mo></msup><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow></mrow></mfrac><msup><mfenced separators="" open="(" close=")"><mi>y</mi><mo>(</mo><mi>φ</mi><mo>(</mo><mi>s</mi><mo>)</mo><mo>)</mo></mfenced><mo>Δ</mo></msup></mfenced></mfenced><mo>Δ</mo></msup><mo>+</mo><mi>η</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>Φ</mo><mrow><mo>(</mo><mi>y</mi><mrow><mo>(</mo><mi>τ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>,</mo><mi>s</mi><mo>∈</mo><msub><mrow><mo>[</mo><msub><mi>s</mi><mn>0</mn></msub><mo>,</mo><mo>∞</mo><mo>)</mo></mrow><mi mathvariant="double-struck">T</mi></msub><mo>.</mo></mrow></semantics></math></inline-formula> By employing generalized Riccati transformation and inequality scaling technique, we establish some oscillation criteria.