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On Eventually Positive Solutions of Quasilinear Second‐Order Neutral Differential Equations

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  • Simona Fišnarová
  • Robert Mařík

Abstract

We study the second‐order neutral delay differential equation [r(t)Φγ(z′(t))]′+q(t)Φβ(x(σ(t)))=0, where Φα(t) = |t|α−1t, α ≥ 1 and z(t) = x(t) + p(t)x(τ(t)). Based on the conversion into a certain first‐order delay differential equation we provide sufficient conditions for nonexistence of eventually positive solutions of two different types. We cover both cases of convergent and divergent integral ∫ ∞ r-1/γ(t)dt. A suitable combination of our results yields new oscillation criteria for this equation. Examples are shown to exhibit that our results improve related results published recently by several authors. The results are new even in the linear case.

Suggested Citation

  • Simona Fišnarová & Robert Mařík, 2014. "On Eventually Positive Solutions of Quasilinear Second‐Order Neutral Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:818732
    DOI: 10.1155/2014/818732
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    References listed on IDEAS

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    1. Shurong Sun & Tongxing Li & Zhenlai Han & Yibing Sun, 2011. "Oscillation of Second-Order Neutral Functional Differential Equations with Mixed Nonlinearities," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-15, January.
    2. Shuhong Tang & Tongxing Li & Ethiraju Thandapani, 2011. "Oscillation Theorems for Second-Order Half-Linear Advanced Dynamic Equations on Time Scales," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-16, September.
    3. Tongxing Li & Zhenlai Han & Chenghui Zhang & Hua Li, 2011. "Oscillation Criteria for Second‐Order Superlinear Neutral Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    4. Shurong Sun & Tongxing Li & Zhenlai Han & Hua Li, 2012. "Oscillation Theorems for Second‐Order Quasilinear Neutral Functional Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    5. Tongxing Li & Zhenlai Han & Chenghui Zhang & Hua Li, 2011. "Oscillation Criteria for Second-Order Superlinear Neutral Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2011, pages 1-17, March.
    6. Ravi P. Agarwal & Said R. Grace & Donal O’Regan, 2000. "Oscillation Theory for Difference and Functional Differential Equations," Springer Books, Springer, number 978-94-015-9401-1, October.
    7. Shurong Sun & Tongxing Li & Zhenlai Han & Yibing Sun, 2011. "Oscillation of Second‐Order Neutral Functional Differential Equations with Mixed Nonlinearities," Abstract and Applied Analysis, John Wiley & Sons, vol. 2011(1).
    8. Shurong Sun & Tongxing Li & Zhenlai Han & Hua Li, 2012. "Oscillation Theorems for Second-Order Quasilinear Neutral Functional Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-17, July.
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