On the oscillatory behaviour of a second order nonlinear differential equation

  • 58 Accesses

  • 13 Citations


The equation y″(t)+p(t)f(y(t))=0 is considered, in which f(y) has the same sign as y, is monotone increasing, and is « strongly nonlinear », and p(t) is locally integrable suchthat \(\mathop {\lim }\limits_{T \to \infty } \mathop \smallint \limits_t^T p(s)ds\) exists. No sign condition is assumed, either on p, or its integral. Under conditions which guarantee that, in a certain sense, p is not too negative, an integrability condition is obtained which is necessary and sufficient for all extendable solutions to oscillate. In the second half of the paper, the monotonicity and strong nonlinearity assumptions on f are shown to be essential ingredients for the existence of such osoillatory criteria. Some examples illustrating the theorems are given.


  1. [1]

    F. V. Atkinson,On second order nonlinear oscillation, Pacific J. Math.,5 (1955), pp. 643–647.

  2. [2]

    T. BurtonR. Grimmer,On continuability of solutions of second order differential equations, Proc. Amer. Math. Soc.,29 (1971), pp. 277–283.

  3. [3]

    W. J. Coles,Oscillation criteria for nonlinear second order equations, Ann. di Mat. Pura ed Appl.,82 (4) (1969), pp. 123–134.

  4. [4]

    L. H. Erbe,Oscillation theorems for second order nonlinear differential equations, Proc. Amer. Math. Soc.,24 (4) (1970), pp. 811–814.

  5. [5]

    L. K. Jackson,Subfunctions and second-order ordinary differential inequalities, Advances in Math.,2 (1968), pp. 307–363.

  6. [6]

    I. T. Kiguradze,A note on the oscillation of u″+a(t)|t| n sgnu=0, Casopis Pest. Math.,92 (1967), pp. 343–350 (Russian).

  7. [7]

    J. W. Macki,An example in the theory of nonlinear oscillations, SIAM J. Appl. Math.,17 (3) (1969), pp. 517–519.

  8. [8]

    J. W. MackiJ. S. W. Wong,Oscillation of solutions to second-order nonlinear differential equations, Pacific J. Math.,24 (1) (1968), pp. 111–117.

  9. [9]

    R. A. MooreZ. Nehari,Non-oscillation theorems for a class of nonlinear differential equations, Trans. Amer. Math. Soc.,93 (1959), pp. 30–52.

  10. [10]

    Z. Nehari,On a class of nonlinear second order differential equations, Trans. Amer. Math. Soc.,95 (1960), pp. 101–123.

  11. [11]

    W. R. Utz,Properties of solutions of u″+g(t)u 2n−1=0, Monatsh. Math.,66 (1962), pp. 55–60.

  12. [12]

    W. R. Utz,Properties of solutions of u″+g(t)u 2n−1=0 (II), Monatsh. Math.,69 (1965), pp. 353–361.

  13. [13]

    P. Waltman,Some properties of u″+a(t)f(u)=0, Monatsh. Math.,67 (1963), pp. 50–54.

  14. [14]

    P. Waltman,An oscillation criterion for a nonlinear second order equation, J. Math. Anal. Appl.,10 (1965), pp. 439–441.

  15. [15]

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

  16. [16]

    J. S. W. Wong,A note on second order nonlinear oscillation, SIAM Review,10 (1968), pp. 88–91.

  17. [17]

    J. S. W. Wong,On second order nonlinear oscillation, Funke. Ekv.,11 (1968), pp. 207–234.

Download references

Author information

Additional information

Entrata in Redazione l’8 agosto 1973.

Research supported in part by NRC grant no. A-8130.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Butler, G.J. On the oscillatory behaviour of a second order nonlinear differential equation. Annali di Matematica 105, 73–92 (1975).

Download citation


  • Differential Equation
  • Integrability Condition
  • Oscillatory Behaviour
  • Nonlinear Differential Equation
  • Essential Ingredient