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Oscillation and Nonoscillation of Half-Linear Differential Equations

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Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations

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Abstract

In this chapter we shall present oscillation and nonoscillation criteria for second order half-linear differential equations. In recent years these equations have attracted considerable attention. This is largely due to the fact that half-linear differential equations occur in a variety of real world problems; moreover, these are the natural generalizations of second order linear differential equations. In Section 3.1, we shall provide some preliminaries for the study of half-linear differential equations. In Sections 3.2 and 3.3, respectively, Sturm’s and Levin’s type comparison theorems are developed. In Section 3.4, we shall establish a Liapunov type inequality. Section 3.5 presents an oscillation criterion for almost periodic Sturm-Liouville equations. A systematic study on the zeros of solutions of singular half-linear equations is made in Section 3.6. Nonoscillation characterizations (necessary and sufficient conditions), comparison results as well as several sufficient criteria for the nonoscillation are presented in Section 3.7. Section 3.8 is devoted to the study of oscillation of half-linear equations. In Section 3.9, we shall establish oscillation criteria by employing integral and weighted averaging techniques. Here, interval criteria for the oscillation of half-linear equations are also provided. Section 3.10 deals with the oscillation of half-linear equations with integrable coefficients. Section 3.11 addresses the oscillation of damped and forced equations. In Section 3.12, we shall derive lower bounds for the distance between consecutive zeros of an oscillatory solution. Finally, in Section 3.13, we shall present a systematic study of the oscillation and nonoscillation of half-linear equations with a deviating argument. Here, classifications of the nonoscillatory solutions, and the existence results which guarantee that the solutions have prescribed asymptotic behavior are also presented.

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References

  1. R.P. Agarwal and S.R. Grace, Oscillations of forced functional differential equations generated by advanced arguments, Aequationes Nlathematicae, to appear.

    Google Scholar 

  2. R.P. Agarwal and S.R. Grace, Oscillation criteria for second order half—linear differential equations with deviating arguments, Dyn. Cont. Disc. Impul. Sys.,to appear.

    Google Scholar 

  3. R.P. Agarwal and S.R. Grace, Oscillation of certain second order differential equations Proc. Sixth Int. Conf. on Nonlinear Functional Analysis and ApplicationsKorea 2000, Nova Science Publishers Inc. New York, to appear.

    Google Scholar 

  4. R.P. Agarwal and S.R. Grace, Interval criteria for oscillation of second order half—linear ordinary differential equations Functional Differential Equationsto appear.

    Google Scholar 

  5. R.P. Agarwal and S.R. Grace, Second order nonlinear forced oscillations, Dyn. Sys. Appl.,to appear.

    Google Scholar 

  6. R.P. Agarwal, S.R. Grace and D. O’Regan, Oscillation criteria for certain nth order differential equations with deviating arguments, J. Math. Anal. Appl.,to appear.

    Google Scholar 

  7. R.P. Agarwal, S.R. Grace and D. O’Regan, On the oscillation of second order functional differential equations, Advances Math]. Sci. Appl.,to appear.

    Google Scholar 

  8. R.P. Agarwal, W.-C. Lian and C.C. Yeh, Levin’s comparison theorems for nonlinear second order differential equations, Applied Math. Letters 9 (1996), 29–35.

    Article  MathSciNet  MATH  Google Scholar 

  9. A. Besicovitch, Almost Periodic Functions, Dover, New York, 1954.

    Google Scholar 

  10. I. Bihari, An oscillation theorem concerning the half-linear differential equations of second order, Magyar Tud. Aka.d. Mat. Kutato Int. Kozl. 8 (1963), 275–280.

    Google Scholar 

  11. I. Bihari, Oscillation and monotonicity theorems concerning nonlinear differential equations of the second order, Acta Math. Sci. Hungar. 9 (1968), 83–104.

    MathSciNet  Google Scholar 

  12. T.A. Chanturia, N. Kandelaki and A. Lomtatidze, On zeros of solutions of a second order singular half-linear equation, Mem. Diff. Eqns. Math. Phy 17 (1999), 127–154.

    Google Scholar 

  13. O. Dosly, Oscillation criteria for half-linear second order differential equations, Hiroshima Math. J. 28 (1998), 507–521.

    MathSciNet  MATH  Google Scholar 

  14. A. Dzurnak and A.B. Mingarelli, Sturm-Liouville equations with Besicovitch almost periodicity, Proc. Amer. Math. Soc. 106 (1989), 647–653.

    MathSciNet  MATH  Google Scholar 

  15. A. Elbert, A half-linear second order differential equation, in Qualitative Theory of Differential equations, Szeged-Societas Janos Bolyai, Colloq. Math. Soc. Janos Bolyai 30, 1979, 153–180.

    MathSciNet  Google Scholar 

  16. A. Elbert, Oscillation and nonoscillation theorems for some nonlinear ordinary differential equations, Lecture Notes in Math., 964, Springer-Verlag, New York, 1982, 187–212.

    Google Scholar 

  17. A. Elbert, Asymptotic behavior of autonomous half-linear differential systems on the plane, Studia Sci. Math. Hungar. 19 (1984), 447–464.

    MathSciNet  MATH  Google Scholar 

  18. A. Elbert, On the half-linear second order differential equations, Acta. Math. Hungar. 49 (1987), 487–508.

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Elbert and T. Kusano, Oscillation and nonoscillation theorems for a class of second order quasilinear differential equations, Acta. Math. Hun-gar. 56 (1990), 325–336.

    Article  MathSciNet  MATH  Google Scholar 

  20. L.H. Erbe, Oscillation criteria for second order nonlinear delay equations, Canad. Math. Bull. 16 (1973), 49–56.

    Article  MathSciNet  MATH  Google Scholar 

  21. S.R. Grace, Oscillation theorems for nonlinear differential equations of second order, J. Math. Anal. Appl. 171 (1992), 220–241.

    Article  MathSciNet  MATH  Google Scholar 

  22. S.R. Grace and B.S. Lalli, Oscillatory behavior for nonlinear second order functional differential equations with deviating arguments, Bull. Inst. Math. Aad. Sinica 14 (1986), 187–196.

    MathSciNet  MATH  Google Scholar 

  23. G.H. Hardy, J.E. Littlewood and G. Polya, Inequalities, 2nd ed. Cambridge Univ. Press, 1988.

    Google Scholar 

  24. B.J. Harris and Q. Kong, On the oscillation of differential equations with an oscillatory coefficient, Trans. Amer. Math. Soc. 347 (1995), 1831 1839.

    MathSciNet  Google Scholar 

  25. P. Hartman, Ordinary Differential Equations, John Wiley, New York, 1964.

    Google Scholar 

  26. W. Jingfa, On second order quasilinear oscillations, Funkcial. Ekvac. 41 (1998), 25–54.

    MathSciNet  MATH  Google Scholar 

  27. A.G. Kartsatos, On oscillations of nonlinear equations of second order, J. Math. Anal. Appl. 24 (1968), 357–361.

    Article  MathSciNet  Google Scholar 

  28. A.G. Kartsatos, On nth order differential inequalities, J. Math. Anal. Appl. 52 (1975), 1–9.

    Article  MathSciNet  MATH  Google Scholar 

  29. Q. Kong, Interval criteria for oscillation of second order linear ordinary differential equations, J. Math. Anal. App]. 239 (1999), 258–270.

    Article  Google Scholar 

  30. R.G. Koplatadze and T.A. Chanturia, On oscillatory and monotone solutions of first order differential equations with deviating arguments, Differencial’nye Uravnenija 18 (1982), 1463–1465.

    MATH  Google Scholar 

  31. T. Kusano and B.S. Lalli, On oscillation of half-linear functional differential equations with deviating arguments, Hiroshima Math. J. 24 (1994), 549–563.

    MathSciNet  MATH  Google Scholar 

  32. T. Kusano, A. Ogata and H. Usami, Oscillation theory for a class of second order quasilinear ordinary differential equations with application to partial differential equations, Japan J. Math. 19 (1993), 131–147.

    MathSciNet  MATH  Google Scholar 

  33. T. Kusano, A. Ogata and H. Usami, On the oscillation of solutions of second order quasilinear ordinary differential equations, Hiroshima Math. J. 23 (1993), 645–667.

    MathSciNet  MATH  Google Scholar 

  34. T. Kusano and Y. Naito, Oscillation and nonoscillation criteria for second order quasilinear differential equations, Acta Math. Hungar. 76 (1997), 81–99.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Kusano, Y. Naito and A. Ogata, Strong condition and nonoscillation of quasilinear differential equations of second order, Diff. Eqns. Dyn. Sys. 2 (1994), 1–10.

    MathSciNet  MATH  Google Scholar 

  36. 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.

    Article  MathSciNet  MATH  Google Scholar 

  37. W. Leighton, The detection of the oscillation of solutions of a second order linear differential equation, Duke J. Math. 17 (1950), 57–61.

    MathSciNet  MATH  Google Scholar 

  38. A.Y. Levin, A comparison principle for second order differential equations, Sov. Math. Dokl. 1 (1960), 1313–1316.

    MATH  Google Scholar 

  39. H.J. Li, Nonoscillation characterization of second order linear differential equations, Math. Nachr. 219 (2000), 147–161.

    Article  MathSciNet  MATH  Google Scholar 

  40. H.J. Li and C.C. Yeh, Sturmian comparison theorem for half linear second order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1193–1204.

    Article  MathSciNet  MATH  Google Scholar 

  41. H.J. Li and C.C. Yeh, Nonoscillation criteria for second order half linear differential equations, App]. Math. Letters 8 (1995), 63–70.

    Article  MathSciNet  MATH  Google Scholar 

  42. H.J. Li and C.C. Yeh, Nonoscillation theorems for second order quasi-linear differential equations, Publ. Math. (Debrecen) 47 (1995), 271–279.

    MathSciNet  MATH  Google Scholar 

  43. H.J. Li and C.C. Yeh, Oscillation of half linear second order differential equations, Hiroshima Math. J. 25 (1995), 585–594.

    MathSciNet  MATH  Google Scholar 

  44. H.J. Li and C.C. Yeh, Oscillation criteria for nonlinear differential equations, Houston J. Math. 21 (1995), 801 811.

    MathSciNet  Google Scholar 

  45. H.J. Li and C.C. Yeh, An oscillation criterion of almost periodic Sturm-Liouville equations, Rocky Mount..1. Math. 25 (1995), 1417–1429.

    MathSciNet  MATH  Google Scholar 

  46. H.J. Li and C.C. Yeh, On the nonoscillatory behavior of solutions of a second order linear differential equation, Math. Nachr. 182(1996), 295315.

    Google Scholar 

  47. H.J. Li and C.C. Yeh, Oscillation of nonlinear functional differential equations of the second order, Appl. Math. Letters 11 (1) (1998), 71–77.

    Article  MathSciNet  MATH  Google Scholar 

  48. H.J. Li and C.C. Yeh, Oscillation and nonoscillation criteria for second order linear differential equations, Math. Nachr. 194 (1998), 171–184.

    Article  MathSciNet  MATH  Google Scholar 

  49. W.C. Lian, C.C. Yeh and H.J. Li, The distance between zeros of an oscillatory solution to a half linear differential equation, Comput. Math. Appl. 29 (1995), 39–43.

    Article  MathSciNet  MATH  Google Scholar 

  50. W.E. Mahfoud, Oscillation and asymptotic behavior of solutions of nth order nonlinear delay differential equations, J. Differential Equations 24 (1977), 75–98.

    Article  MathSciNet  MATH  Google Scholar 

  51. W.E. Mahfoud, Comparison theorems for delay differential equations, Pacific J. Math. 83 (1979), 187–197.

    Article  MathSciNet  MATH  Google Scholar 

  52. W.E. Mahfoud and S.M. Rankin, Some properties of solutions of (r(t)T(x)x’)’ + a(t) f (x) = 0, SIAM J. Math, Anal. 10 (1979), 49–54.

    MathSciNet  MATH  Google Scholar 

  53. J.V. Manojlovic, Oscillation criteria for a second order half-linear differential equation, Math. Comput. Modelling 30 (5–6) (1999), 109–119.

    Article  MathSciNet  MATH  Google Scholar 

  54. Ch.G. Philos, On the existence of nonoscillatory solutions tending to zero at oo for differential equations with positive delays, Arch. Math. 36 (1981), 168–178.

    MathSciNet  MATH  Google Scholar 

  55. Ch.G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. 53 (1989), 482–492.

    MathSciNet  MATH  Google Scholar 

  56. M. Del Pino and R. Manasevich, Oscillation and nonoscillation for Ou’IP-2u’)ß + a(t)luIP-2u = 0, p 1, Houston J. Math. 14(1988), 173177.

    Google Scholar 

  57. B. Singh, Comparative study of asymptotic nonoscillation and quickly oscillation of second order linear differential equations, J. Math]. Phyl. Sci. 4 (1974), 363–376.

    Google Scholar 

  58. C.A. Swanson, Comparison and Oscillation Theory of Linear Differential Equations, Academic Press, New York, 1968.

    Google Scholar 

  59. D. Willett, On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175–194.

    MathSciNet  MATH  Google Scholar 

  60. A. Wintner, A criterion for oscillatory stability, Quart. Appl. Math. 7 (1949), 115–117.

    MathSciNet  MATH  Google Scholar 

  61. A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951), 368–380.

    Article  MathSciNet  MATH  Google Scholar 

  62. F.H. Wong and C.C. Yeh, An oscillation criterion for Sturm—Liouville equations with Besicovitch almost periodic coefficients, Hiroshima Math. J. 21 (1991), 521–528.

    MathSciNet  MATH  Google Scholar 

  63. J.S.W. Wong, Oscillation criteria for a forced second order linear differential equation, J. Math. Anal. Appl. 231 (1999), 235–240.

    Article  MathSciNet  MATH  Google Scholar 

  64. P.J.Y. Wong and R.P. Agarwal, Oscillation theorems and existence criteria of asymptotically monotone solutions for second order differential equations, Dynam. Systems Appl. 4 (1995), 477–496.

    MathSciNet  MATH  Google Scholar 

  65. P.J.Y. Wong and R.P. Agarwal, On the oscillation and asymptotically monotone solutions of second order quasilinear differential equations, Appl. Math. Comput. 79 (1996), 207–237.

    MathSciNet  MATH  Google Scholar 

  66. P.J.Y. Wong and R.P. Agarwal, Oscillatory behavior of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl. 198 (1996), 813–829.

    Article  Google Scholar 

  67. P.J.Y. Wong and R.P. Agarwal, Oscillation criteria for half -linear differential equations, Advances Math. Sci. App]. 9 (1999), 649–663.

    MathSciNet  MATH  Google Scholar 

  68. J. Yan, Oscillation property for second order differential equations with an `integral small’ coefficient, Acta Math. Sinica 30 (1987), 206–215.

    MATH  Google Scholar 

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Authors and Affiliations

  1. Florida Institute of Technology, Melbourne, Florida, USA

    Ravi P. Agarwal

  2. Cairo University, Orman, Giza, Egypt

    Said R. Grace

  3. National University of Ireland, Galway, Ireland

    Donal O’Regan

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  1. Ravi P. Agarwal
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  2. Said R. Grace
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  3. Donal O’Regan
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Agarwal, R.P., Grace, S.R., O’Regan, D. (2002). Oscillation and Nonoscillation of Half-Linear Differential Equations. In: Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-2515-6_3

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  • DOI: https://doi.org/10.1007/978-94-017-2515-6_3

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