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Zbl 1027.34039
Ou, C.H.; Wong, James S.W.
On existence of oscillatory solutions of second order Emden-Fowler equations.
(English)
[J] J. Math. Anal. Appl. 277, No.2, 670-680 (2003). ISSN 0022-247X

The authors study the second-order Emden-Fowler equation $$y''+a|y|^\gamma \text{sgn} y=0, \quad \gamma>0,\ \gamma\ne 1,\tag 1$$ where $a$ is a positive absolutely continuous function on $(0, \infty)$. Let $\phi$ be the function defined by $\phi(x)=a(x) x^{\frac{\gamma+3}{2}}$, and assume that $\phi$ is bounded away from zero at infinity. Under this condition, the main result of the paper says that if the negative part of $\phi'$, $\phi_{-}'(x)=-\min(\phi'(x), 0)$, belongs to $L^1(0, \infty)$, 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 $\phi$, strictly extends previous results of {\it M. Jasny} [Cas. Pest. Mat. 85, 78-82 (1960; Zbl 0113.07603)], {\it J. Kurzweil} [Cas. Pest. Mat. 85, 357-358 (1960; Zbl 0129.06204)], {\it J. W. Heidel} and {\it D. B. Hinton} [SIAM J. Math. Anal. 3, 344-351 (1972; Zbl 0243.34062)], {\it L. H. Erbe} and {\it J. S. Muldowney} [Ann. Mat. Pura Appl., IV. Ser. 109, 23-38 (1976; Zbl 0345.34022)], and {\it K. Chiou} [Proc. Am. Math. Soc. 35, 120-122 (1972; Zbl 0262.34026)].
[Ingrid Beltita (Bucureşti)]
MSC 2000:
*34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A34 Nonlinear ODE and systems, general

Keywords: oscillatory solution; Emden-Fowler equation

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