Ukrainian Mathematical Journal

, Volume 58, Issue 12, pp 1935–1949 | Cite as

Multilayer structures of second-order linear differential equations of Euler type and their application to nonlinear oscillations

  • N. Yamaoka
  • J. Sugie
Article
  • 38 Downloads

Abstract

The purpose of this paper is to present new oscillation theorems and nonoscillation theorems for the nonlinear Euler differential equation t2x″ + g(x) = 0. Here we assume that xg(x) > 0 if x ≠ 0, but we do not necessarily require that g(x) be monotone increasing. The obtained results are best possible in a certain sense. To establish our results, we use Sturm’s comparison theorem for linear Euler differential equations and phase plane analysis for a nonlinear system of Liénard type.

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References

  1. 1.
    P. Hartman, “On the linear logarithmico-exponential differential equation of the second order,” Amer. J. Math., 70, 764–779 (1948).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Hille, “Non-oscillation theorems,” Trans. Amer. Math. Soc., 64, 234–252 (1948).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    J. C. P. Miller, “On a criterion for oscillatory solutions of a linear differential equation of the second order,” Proc. Cambridge Phil. Soc., 36, 283–287 (1940).MATHCrossRefGoogle Scholar
  4. 4.
    D. Willett, “Classification of second order linear differential equations with respect to oscillation,” Adv. Math., 3, 594–623 (1969).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. Kneser, “Untersuchungen über die reelen Nullstellen der Integrale linearer Differentialgleichungen,” Math. Ann., 42, 409–435 (1893).CrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Kneser, “Untersuchung und asymptotische Darstellung der Integrale gewisser Differentialgleichungen bei grossen reellen Werthen des Arguments,” J. Reine Angew. Math., 116, 178–212 (1896).Google Scholar
  7. 7.
    C. A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York (1968).MATHGoogle Scholar
  8. 8.
    J. Sugie, “Oscillation criteria of Kneser-Hille type for second-order differential equations with nonlinear perturbed terms,” Rocky Mountain J. Math., 34, 1519–1537 (2004).MATHMathSciNetGoogle Scholar
  9. 9.
    J. Sugie and N. Yamaoka, “An infinite sequence of nonoscillation theorems for second-order nonlinear differential equations of Euler type,” Nonlin. Analysis, 50, 373–388 (2002).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    J. Sugie and N. Yamaoka, “Oscillation of solutions of second-order nonlinear self-adjoint differential equations,” J. Math. Anal. Appl., 291, 387–405 (2004).MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    J. Sugie and T. Hara, “Nonlinear oscillation of second order differential equations of Euler type,” Proc. Amer. Math. Soc., 124, 3173–3181 (1996).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    M. Cecchi, M. Marini, and G. Villari, “On some classes of continuable solutions of a nonlinear differential equation,” J. Different. Equat., 118, 403–419 (1995).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    M. Cecchi, M. Marini, and G. Villari, “Comparison results for oscillation of nonlinear differential equations,” NoDEA Nonlinear Different. Equat. Appl., 6, 173–190 (1999).MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    C.-H. Ou and J. S. W. Wong, “On existence of oscillatory solutions of second order Emden-Fowler equations,” J. Math. Anal. Appl., 277, 670–680 (2003).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    J. S. W. Wong, “Oscillation theorems for second-order nonlinear differential equations of Euler type,” Meth. Appl. Anal., 3, 476–485 (1996).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • N. Yamaoka
    • 1
  • J. Sugie
    • 2
  1. 1.Sophia UniversityTokyoJapan
  2. 2.Shimane UniversityMatsueJapan

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