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On existence of oscillatory solutions of second order Emden-Fowler equations. (English) Zbl 1027.34039

The authors study the second-order Emden-Fowler equation

y '' +a|y| γ sgny=0,γ>0,γ1,(1)

where a is a positive absolutely continuous function on (0,). Let ϕ be the function defined by ϕ(x)=a(x)x γ+3 2 , and assume that ϕ is bounded away from zero at infinity. Under this condition, the main result of the paper says that if the negative part of ϕ ' , ϕ - ' (x)=-min(ϕ ' (x),0), belongs to L 1 (0,), then equation (1) has oscillatory solutions (that is, solutions with arbitrary large zeroes). The authors provide an example that shows that this result, being applicable to nonmonotonous functions ϕ, strictly extends previous results of M. Jasny [Cas. Pest. Mat. 85, 78-82 (1960; Zbl 0113.07603)], J. Kurzweil [Cas. Pest. Mat. 85, 357-358 (1960; Zbl 0129.06204)], J. W. Heidel and D. B. Hinton [SIAM J. Math. Anal. 3, 344-351 (1972; Zbl 0243.34062)], L. H. Erbe and J. S. Muldowney [Ann. Mat. Pura Appl., IV. Ser. 109, 23-38 (1976; Zbl 0345.34022)], and K. Chiou [Proc. Am. Math. Soc. 35, 120-122 (1972; Zbl 0262.34026)].

34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A34Nonlinear ODE and systems, general