Opuscula Math. 31, no. 3 (2011), 373-391
http://dx.doi.org/10.7494/OpMath.2011.31.3.373

Opuscula Mathematica

# Oscillation theorems concerning non-linear differential equations of the second order

E. M. Elabbasy
Sh. R. Elzeiny

Abstract. This paper concerns the oscillation of solutions of the differential eq. $\left[ r\left( t\right) \psi \left(x\left( t\right) \right) f\text{ }( \overset{\cdot }{x}(t))\right]^{\cdot }+q\left( t\right) \varphi (g\left( x\left( t\right) \right), r\left( t\right) \psi \left( x\left( t\right) \right) f(\overset{\cdot }{x}(t)))=0,$ where $$u\varphi \left( u,v\right) \gt 0$$ for all $$u\neq 0$$, $$xg\left( x\right) \gt 0$$, $$xf\left( x\right)\gt 0$$ for all $$x\neq 0$$, $$\psi \left( x\right) \gt 0$$ for all $$x\in \mathbb{R}$$, $$r\left( t\right) \gt 0$$ for $$t\geq t_{0}\gt 0$$ and $$q$$ is of arbitrary sign. Our results complement the results in [A.G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 665–668], and improve a number of existing oscillation criteria. Our main results are illustrated with examples.

Keywords: second order, nonlinear, differential equations, oscillation.

Mathematics Subject Classification: 34C10, 34C15.

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• E. M. Elabbasy
• Mansoura University, Department of Mathematics, Faculty of Science, Mansoura 35516, Egypt
• Sh. R. Elzeiny
• Al-Baha University, Department of Mathematics, Faculty of Science, Kingdom of Saudi Arabia, P.O. Box 1034
• Revised: 2010-09-26.
• Accepted: 2010-10-05.