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On the oscillation of third-order quasi-linear neutral functional differential equations

Ethiraju Thandapani, Tongxing Li (2011)

Archivum Mathematicum

The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation [ a ( t ) ( [ x ( t ) + p ( t ) x ( δ ( t ) ) ] ' ' ) α ] ' + q ( t ) x α ( τ ( t ) ) = 0 , E where α > 0 , 0 p ( t ) p 0 < and δ ( t ) t . By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.

On unstable neutral differential equations of the second order

Jozef Džurina (2002)

Czechoslovak Mathematical Journal

The aim of this paper is to present sufficient conditions for all bounded solutions of the second order neutral differential equation ( x ( t ) - p x ( t - τ ) ) ' ' - q ( t ) x ( σ ( t ) ) = 0 to be oscillatory and to improve some existing results. The main results are based on the comparison principles.

Oscillation conditions for first-order nonlinear advanced differential equations

Özkan Öcalan, Nurten Kiliç (2023)

Commentationes Mathematicae Universitatis Carolinae

Our purpose is to analyze a first order nonlinear differential equation with advanced arguments. Then, some sufficient conditions for the oscillatory solutions of this equation are presented. Our results essentially improve two conditions in the paper “Oscillation tests for nonlinear differential equations with several nonmonotone advanced arguments” by N. Kilıç, Ö. Öcalan and U. M. Özkan. Also we give an example to illustrate our results.

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