Opuscula Math. 35, no. 4 (2015), 485-497

http://dx.doi.org/10.7494/OpMath.2015.35.4.485

Opuscula Mathematica

# Oscillation criteria for third order nonlinear delay differential equations with damping

Abstract. This note is concerned with the oscillation of third order nonlinear delay differential equations of the form \[\left( r_{2}(t)\left( r_{1}(t)y^{\prime}(t)\right)^{\prime}\right)^{\prime}+p(t)y^{\prime}(t)+q(t)f(y(g(t)))=0.\tag{\(\ast\)}\] In the papers [A. Tiryaki, M. F. Aktas, Oscillation criteria of a certain class of third order nonlinear delay differential equations with damping, J. Math. Anal. Appl. 325 (2007), 54-68] and [M. F. Aktas, A. Tiryaki, A. Zafer, Oscillation criteria for third order nonlinear functional differential equations, Applied Math. Letters 23 (2010), 756-762], the authors established some sufficient conditions which insure that any solution of equation \((\ast)\) oscillates or converges to zero, provided that the second order equation \[\left( r_{2}(t)z^{\prime }(t)\right)^{\prime}+\left(p(t)/r_{1}(t)\right) z(t)=0\tag{\(\ast\ast\)}\] is nonoscillatory. Here, we shall improve and unify the results given in the above mentioned papers and present some new sufficient conditions which insure that any solution of equation \((\ast)\) oscillates if equation \((\ast\ast)\) is nonoscillatory. We also establish results for the oscillation of equation \((\ast)\) when equation \((\ast\ast)\) is oscillatory.

Keywords: oscillation, third order, delay differential equation.

Mathematics Subject Classification: 34C10, 39A10.

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- Said R. Grace
- Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12613, Egypt

- Communicated by P.A. Cojuhari.

- Received: 2014-04-24.
- Revised: 2014-10-12.
- Accepted: 2014-10-28.
- Published online: 2015-02-06.