Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
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OSCILLATION THEOREMS FOR CERTAIN SECOND ORDER NONLINEAR DIFFERENTIAL EQUATIONS

  • Sun, Yibing (School of Science, University of Jinan) ;
  • Han, Zhenlai (School of Science, University of Jinan, School of Control Science and Engineering, Shandong University) ;
  • Zhao, Ping (School of Control Science and Engineering, University of Jinan) ;
  • Sun, Ying (School of Science, University of Jinan)
  • Received : 2010.10.12
  • Accepted : 2011.01.10
  • Published : 2011.09.30
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Abstract

In this paper, we consider the oscillation of the following certain second order nonlinear differential equations $(r(t)(x^{\prime}(t))^{\alpha})^{\prime}+q(t)x^{\beta}(t)=0$>, where ${\alpha}$ and ${\beta}$ are ratios of positive odd integers. New oscillation theorems are established, which are based on a class of new functions ${\Phi}={\Phi}(t,s,l)$ defined in the sequel. Also, we establish some interval oscillation criteria for this equation.

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References

  1. A. Elbert, A half-linear second order differential equation, Colloquia Math. soc. 30 (1979), 153-180.
  2. A. Elbert, Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, in: Ordinary and Partial Differential for Equations, Lecture Notes in Mathematics. 964 (1982), 187-212.
  3. S. R. Grace, Oscillation theorems nonlinear differential equations of second order, J. Math. Anal. Appl. 171 (1992), 220-241. https://doi.org/10.1016/0022-247X(92)90386-R
  4. T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81-99. https://doi.org/10.1007/BF02907054
  5. T. Kusano, Y. Naito and A. Ogata, Strong ocillation and nonoscillation of quasilinear differential equations of second order, Diff. Eqs. Dyn. Syst. 2 (1994), 1-10.
  6. T. Kusano and N. Yoshida, Nonoscillation theorems for a class of quasilinear differential equations of second order, J. Math. Anal. Appl. 189 (1995), 115-127. https://doi.org/10.1006/jmaa.1995.1007
  7. J. Lagnese and Y. V. Rogovchenko, Oscillation criteria for certain nonlinear differential equations, J. Math. Anal. Appl. 229 (1999), 399-416. https://doi.org/10.1006/jmaa.1998.6148
  8. D. D. Mirsov, On some analogs of Sturm's and Kneser's theorems for nonlinear systems, J. Math. Anal. Appl. 53 (1976), 418-425. https://doi.org/10.1016/0022-247X(76)90120-7
  9. D. D. Mirsov, On the oscillation of solutions of a system of differential equations, Math. Zametki. 23 (1978), 401-404.
  10. Z. Han, T. Li, S. Sun and Y. Sun, Remarks on the paper [Appl. Math. Comput. 207 (2009) 388-396], Appl. Math. Comput. 215 (2010), 3998-4007. https://doi.org/10.1016/j.amc.2009.12.006
  11. I. V. Kamenev, An integral criteria for oscillation of linear differential equations of second order, Mat. Zametki 23 (1978), 249-251.
  12. Q. Long and Q. R. Wang, New oscillation criteria of second-order nonlinear differential equations, Appl. Math. Comput. 212 (2009), 357-365. https://doi.org/10.1016/j.amc.2009.02.040
  13. Q. Kong, Interval criteria for oscillation of second-order linear ordinary differential equa- tions, J. Math. Anal. Appl. 229 (1999), 258-270. https://doi.org/10.1006/jmaa.1998.6159
  14. W. T. Li and R. P. Agarwal, Interval oscillation criteria for second-order nonlinear differential equations with damping, Comput. Math. Appl. 40 (2000), 217-230. https://doi.org/10.1016/S0898-1221(00)00155-3
  15. W. T. Li and R. P. Agarwal, Interval oscillation criteria related to integral averaging technique for certain nonlinear differential equations, J. Math. Anal. Appl. 245 (2000), 171-188. https://doi.org/10.1006/jmaa.2000.6749
  16. Q. R. Wang, Interval criteria for oscillation of certain second order nonlinear differential equations, Dynam. Cont. Discr. Impul. Syst. Ser A: Math. Anal. 12 (2005), 769-781.