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
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.
- 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
- Received: 2010-05-27.
- Revised: 2010-09-26.
- Accepted: 2010-10-05.
