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Literatur
W. E. Milne, A theorem of oscillation,Bull. Amer. Math. Soc.,28 (1922), pp. 102–104.
In order to prove the oscillating character of the solution, it is sufficient to assume instead of “f(y) is odd” that sgf(y)=sgy.
E. G.h(u) may be an even function decreasing foru≥0.
Similarly, it may readily be seen thatall the solutions of (1) have this character, provided thatb is large enough compared to η.
Iff(y) is like that in footnote2, then the maxima form a decreasing sequence and similarly the minima form another one.
Iff(y) is like that in footnote2, then the maxima are equal and the minima too, but their absolute values may be different.
M. Nagumo, Über die Differentialgleichungy″=f(x, y, y′), Proc. of the Phys.-math. Soc. of Japan (3),19 (1937), pp. 861–865.
See the discussion of this equation in a forthcoming paper of the author.
This is a generalization of a theorem ofE. Makai concerning the linear equationy″+ϕ(x)y=0: On a monotonity property of certain Sturm—Liouville functions,Acta Math. Acad. Sci. Hung.,3 (1952), pp. 165–172.
See the notations in § 1.
VizH(u) andF(u) are increasing functions foru>0 and have also increasing inverse functions.
R. Cooke, A monotonity property of Bessel functions,Journal London Math. Soc.,12 (1937), pp. 180–185.
Loc. cit. in § 7.
Still unpublished.
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Bihari, I. Oscillation and monotonity theorems concerning non-linear differential equations of the second order. Acta Mathematica Academiae Scientiarum Hungaricae 9, 83–104 (1958). https://doi.org/10.1007/BF02023866
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DOI: https://doi.org/10.1007/BF02023866