Oscillation of solutions of certain ordinary differential equations of 𝑛𝑡ℎ order

Ryder, Gerald H.; Wend, David V. V.

doi:10.1090/S0002-9939-1970-0261091-5

Oscillation of solutions of certain ordinary differential equations of $n\textrm {th}$ order

Authors: Gerald H. Ryder and David V. V. Wend
Journal: Proc. Amer. Math. Soc. 25 (1970), 463-469
MSC: Primary 34.42
DOI: https://doi.org/10.1090/S0002-9939-1970-0261091-5
MathSciNet review: 0261091
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Abstract: Necessary and sufficient conditions are given that all solutions of ${y^{(n)}} + f(t, y) = 0$ which are continuable to infinity are oscillatory in the case $n$ is even and are oscillatory or strongly monotone in the case $n$ is odd. The results generalize to arbitrary $n$ recent results of J. Macki and J. S. W. Wong for the case $n = 2$ and include as special cases results of I. Kiguradze, I. Ličko and M. Švec, and Š. Belohorec.

Š. Belohorec, Oscillatory solutions of certain nonlinear differential equations of the second order, Mat.-Fyz. Časopis Sloven. Akad. Vied 11 (1961), 250-255. (Slovak)

I. T. Kiguradze, The capability of certain solutions of ordinary differential equations to oscillate, Dokl. Akad. Nauk SSSR 144 (1962), 33–36 (Russian). MR 0136817

---, The problem of oscillations of solutions of nonlinear differential equations, J. Differential Equations 3 (1967), 773-782.

Imrich Ličko and Marko Švec, Le caractère oscillatoire des solutions de l’équation $y^{(n)}+f(x)y^{\alpha }=0,\,n>1$, Czechoslovak Math. J. 13(88) (1963), 481–491 (French, with Russian summary). MR 161001
Adolf Kneser, Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen, Math. Ann. 42 (1893), no. 3, 409–435 (German). MR 1510784, DOI https://doi.org/10.1007/BF01444165
Jack W. Macki and James S. W. Wong, Oscillation of solutions to second-order nonlinear differential equations, Pacific J. Math. 24 (1968), 111–117. MR 224908