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Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Arun K. Tripathy, Shyam S. Santra (2021)

Mathematica Bohemica

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type ( r ( t ) ( z ' ( t ) ) γ ) ' + ∑ i = 1 m q i ( t ) x α i ( σ i ( t ) ) = 0 , t ≥ t 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) . Under the assumption ∫ ∞ ( r ( η ) ) - 1 / γ d η = ∞ , we consider two cases when γ > α i and γ < α i . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.

New interval oscillation criteria for second-order functional differential equations with nonlinear damping

Süleyman Öǧrekçi (2015)

Open Mathematics

This paper concerns the oscillation problem of second-order nonlinear damped ODE with functional terms.We give some new interval oscillation criteria which is not only based on constructing a lower solution of a Riccati type equation but also based on constructing an upper solution for corresponding Riccati type equation. We use a recently developed pointwise comparison principle between those lower and upper solutions to obtain our results. Some illustrative examples are also provided to demonstrate...

New oscillation criteria for first order nonlinear delay differential equations

Xianhua Tang, Jianhua Shen (2000)

Colloquium Mathematicae

New oscillation criteria are obtained for all solutions of a class of first order nonlinear delay differential equations. Our results extend and improve the results recently obtained by Li and Kuang [7]. Some examples are given to demonstrate the advantage of our results over those in [7].

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