Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the location of zeros of oscillatory solution


Author: H. Gingold
Journal: Trans. Amer. Math. Soc. 279 (1983), 471-496
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9947-1983-0709564-6
MathSciNet review: 709564
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The location of zeros of solutions of second order singular differential equations is provided by a new asymptotic decomposition formula. The approximate location of zeros is provided with high accuracy error estimates in the neighbourhood of the point at infinity. The same asymptotic formula suggested is applicable to the neighbourhood of most types of singularities as well as to the neighbourhoods of regular points.


References [Enhancements On Off] (What's this?)

  • [1] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955. MR 0069338 (16:1022b)
  • [2] H. Gingold, Asymptotic decompositions and turning points (in preparation).
  • [3] -, Uniqueness of boundary value problems of second order differential systems and equations, J. Math. Anal. Appl. 73 (1980), 392-410. MR 563991 (82d:34038)
  • [4] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [5] E. Hille, Lectures on differential equations, Addison-Wesley, Reading, Mass., 1969. MR 0249698 (40:2939)
  • [6] K. Kreith, Oscillation theory, Lecture Notes in Math., Vol. 324, Springer-Verlag, Berlin, Heidelberg and New York, 1973.
  • [7] V. Komkov, On the location of zeros of second order differential equations, Proc. Amer. Math. Soc. 35 (1972), 217-222. MR 0298128 (45:7180)
  • [8] W. Leighton, Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603-610. MR 0140759 (25:4173)
  • [9] J. W. Macki and G. J. Butler, Oscillations and comparison theorems for second order linear differential equations with integrable coefficients (Proc. of the C. Carathéodory International Symposium, Athens, 1973), Greek Math. Soc., Athens, 1976, pp. 409-413. MR 0473338 (57:13009)
  • [10] Z. Nehari, Oscillation for second order linear differential equations, Trans. Amer. Math. Soc. 85 (1957), 428-445. MR 0087816 (19:415a)
  • [11] F. W. J. Olver, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [12] Y. Sibuya, Global theory of a second order linear ordinary differential equation with a polynomial coefficient, Mathematics Studies 18, North-Holland, Amsterdam, 1975. MR 0486867 (58:6561)
  • [13] C. A. Swanson, Comparison and oscillation theory of linear differential equations, Academic Press, New York, 1968. MR 0463570 (57:3515)
  • [14] W. Wasow, Asymptotic expansions for ordinary differential equations, Interscience, New York, 1965. MR 0203188 (34:3041)
  • [15] D. Willett, Classification of second order ordinary differential equations with respect to oscillation, Adv. in Math. 3 (1969), 594-623. MR 0280800 (43:6519)
  • [16] A. Wiman, Uber die reellen Losungen der linearen Differentialgleichungen zweiter Ordnung, Ark. Mat. Astr. Fry. 12 (1917).
  • [17] E. Kamke, Differentialgleichungen Losungemethoden und Losungen. I, Becker and Erler, Leipzig, 1943 and Edwards, Ann Arbor, Mich., 1945. MR 0025036 (9:587a)
  • [18] W. Leighton, The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950), 57-62. MR 0032065 (11:248c)
  • [19] M. Rab, Kriterien fur Oscillation der Losungen der Differentialgleichug $ (p(x)y^{\prime})^{\prime} + g(x)y = 0$, Časopis Pěst. Mat. 84 (1959), 335-370; Errata, Ibid. (1960), 91. MR 0114964 (22:5773)
  • [20] J. S. W. Wong, A note on second order oscillation, SIAM Rev. 10 (1968), 88-91. MR 0227525 (37:3109)
  • [21] -, Oscillation and non-oscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197-215. MR 0251305 (40:4536)
  • [22] D. Willet, Classification of second order ordinary differential equations with respect to oscillation, Adv. in Math. 3 (1969), 594-623. MR 0280800 (43:6519)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 34C10

Retrieve articles in all journals with MSC: 34C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0709564-6
Keywords: Oscillation, zeros, second order differential equations, oscillatory solution
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society