Oscillation theorems for second order nonlinear differential equations

Wong, James S. W.

doi:10.1090/S0002-9939-1989-0952324-2

Oscillation theorems for second order nonlinear differential equations
HTML articles powered by AMS MathViewer

by James S. W. Wong PDF

Proc. Amer. Math. Soc. 106 (1989), 1069-1077 Request permission

Abstract:

Oscillation criteria for the second-order nonlinear differential equation $x'' + a\left ( t \right ){\left | x \right |^y}\operatorname {sgn} x = 0 \gamma \ne 1$, are studied where the coefficient $a\left ( t \right )$ is not assumed to be non-negative. New proofs are given to theorems of Butler, and extend earlier results of the author.

References

F. V. Atkinson, On second-order non-linear oscillations, Pacific J. Math. 5 (1955), 643–647. MR 72316, DOI 10.2140/pjm.1955.5.643

Oscillatory solutions of certain nonlinear differential equations of the second order

11

G. J. Butler, On the oscillatory behaviour of a second order nonlinear differential equation, Ann. Mat. Pura Appl. (4) 105 (1975), 73–92. MR 425252, DOI 10.1007/BF02414924
G. J. Butler, The existence of continuable solutions of a second order differential equation, Canadian J. Math. 29 (1977), no. 3, 472–479. MR 499370, DOI 10.4153/CJM-1977-051-7
G. J. Butler, Integral averages and the oscillation of second order ordinary differential equations, SIAM J. Math. Anal. 11 (1980), no. 1, 190–200. MR 556509, DOI 10.1137/0511017
W. J. Coles, Oscillation criteria for nonlinear second order equations, Ann. Mat. Pura Appl. (4) 82 (1969), 123–133. MR 255907, DOI 10.1007/BF02410793
William Benjamin Fite, Concerning the zeros of the solutions of certain differential equations, Trans. Amer. Math. Soc. 19 (1918), no. 4, 341–352. MR 1501107, DOI 10.1090/S0002-9947-1918-1501107-2
Philip Hartman, On non-oscillatory linear differential equations of second order, Amer. J. Math. 74 (1952), 389–400. MR 48667, DOI 10.2307/2372004
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
I. V. Kamenev, Certain specifically nonlinear oscillation theorems, Mat. Zametki 10 (1971), 129–134 (Russian). MR 287077
I. V. Kamenev, An integral test for conjugacy for second order linear differential equations, Mat. Zametki 23 (1978), no. 2, 249–251 (Russian). MR 486798
Walter Leighton, The detection of the oscillation of solutions of a second order linear differential equation, Duke Math. J. 17 (1950), 57–61. MR 32065
Paul Waltman, An oscillation criterion for a nonlinear second order equation, J. Math. Anal. Appl. 10 (1965), 439–441. MR 173050, DOI 10.1016/0022-247X(65)90138-1
D. Willett, On the oscillatory behavior of the solutions of second order linear differential equations, Ann. Polon. Math. 21 (1969), 175–194. MR 249723, DOI 10.4064/ap-21-2-175-194
Aurel Wintner, A criterion of oscillatory stability, Quart. Appl. Math. 7 (1949), 115–117. MR 28499, DOI 10.1090/S0033-569X-1949-28499-6
James S. W. Wong, Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients, Trans. Amer. Math. Soc. 144 (1969), 197–215. MR 251305, DOI 10.1090/S0002-9947-1969-0251305-6
James S. W. Wong, A second order nonlinear oscillation theorem, Proc. Amer. Math. Soc. 40 (1973), 487–491. MR 318585, DOI 10.1090/S0002-9939-1973-0318585-6
James S. W. Wong, Oscillation theorems for second order nonlinear differential equations, Bull. Inst. Math. Acad. Sinica 3 (1975), no. 2, 283–309. MR 390372
James S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975), 339–360. MR 367368, DOI 10.1137/1017036
James S. W. Wong, An oscillation criterion for second order nonlinear differential equations, Proc. Amer. Math. Soc. 98 (1986), no. 1, 109–112. MR 848886, DOI 10.1090/S0002-9939-1986-0848886-3