1949
DOI: 10.1090/qam/28499
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A criterion of oscillatory stability

Abstract: 1. In the linear differential equationx" + f{t)x = 0,let f(t) be a real-valued, continuous function for large positive t, say for l" ^ t < oo. Consider only those solutions x{t) of (1) which are real-valued and distinct from the trivial solution (identically zero). Then, if N(t) denotes the number of zeros of x($) on the interval t0 S s tk t, it is clear from Sturm's separation theorem thatcannot hold for a particular solution x{t) unless it holds for every solution x(t). In this case, (1) is called oscillator… Show more

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Cited by 284 publications
(151 citation statements)
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“…Note that Kamenev studied the more general equation The aim of this paper is to study oscillation properties of (1.1) via modified Riccati technique and obtain extensions of Fite-Wintner [5,20] and Kamenev [10] for this equation, thereby improving results of Mařík [13] and Usami [19]. It is emphasized that the oscillation criteria obtained here are new even for (1.2) and (1.3).…”
Section: Introductionmentioning
confidence: 76%
See 1 more Smart Citation
“…Note that Kamenev studied the more general equation The aim of this paper is to study oscillation properties of (1.1) via modified Riccati technique and obtain extensions of Fite-Wintner [5,20] and Kamenev [10] for this equation, thereby improving results of Mařík [13] and Usami [19]. It is emphasized that the oscillation criteria obtained here are new even for (1.2) and (1.3).…”
Section: Introductionmentioning
confidence: 76%
“…As we know, concerning the oscillation of (1.4) there exists well-elaborated theory, and the most important simple oscillation criterion is the well-known FiteWintner theorem [5,20] which states that if c ∈ C([t 0 , ∞), ‫)ޒ‬ and satisfies [7,17]). By constructing function sequences, in 1977 Kamenev [10] gave some beautiful oscillation criteria for (1.4) under the assumption that c(t) is an "integrally small" coefficient, that is,…”
Section: Introductionmentioning
confidence: 99%
“…If we takea = o" = l in Theorem 2, we obtain the well-known theorem of Wintner [4], and if we consider only averaging pairs of the form (a-, 1), we obtain Coles' Theorem. If we restrict ourselves to averaging pairs of the form (1/a, a) with J" (l/a(s))¿5 = + oo, we obtain extensions of results due to Howard.…”
mentioning
confidence: 92%
“…In the previous two decades, there has been increasing interest in obtaining sufficient conditions for the oscillation and non-oscillation of solutions of different classes of second order differential equations, see for example [2][3][4][5][6][7][8][9][10][11][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] and the references therein.…”
Section: Introductionmentioning
confidence: 99%