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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 $$y''+a|y|^\gamma \text{sgn} y=0, \quad \gamma>0,\ \gamma\ne 1,\tag 1$$ 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)].

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A34 Nonlinear ODE and systems, general
##### Keywords:
oscillatory solution; Emden-Fowler equation
Full Text:
##### References:
 [1] Chiou, K. L.: The existence of oscillatory solutions for the equation y”+$q(t)$y${\gamma}=0, 0{\gamma}$1. Proc. amer. Math. soc. 35, 120-122 (1972) · Zbl 0262.34026 [2] Erbe, L. H.; Muldowney, J. S.: On the existence of oscillatory solutions to nonlinear differential equations. Ann. mat. Pure appl. 109, 23-38 (1976) · Zbl 0345.34022 [3] Heidel, J. W.: Uniqueness, continuation and nonoscillation for a second order nonlinear differential equation. Pacific J. Math. 32, 715-721 (1970) · Zbl 0188.14301 [4] Heidel, J. W.; Hinton, D. B.: Existence of oscillatory solutions for a nonlinear differential equation. SIAM J. Math. anal. 3, 344-351 (1972) · Zbl 0243.34062 [5] Jansy, M.: On the existence of an oscillatory solution of nonlinear differential equation of second order y”$(x)+f(x)$y2n-1=0, $f(x)$0. Ca\check{}sopis pěst mat. 85, 73-83 (1960) [6] Kiguradze, I. T.: A note on the oscillation of solution of equation u”+$a(t)$|u|nsgnu=0. Ca\check{}sopis pěst mat. 92, 343-350 (1967) [7] Kurzweil, J.: A note on oscillatory solutions of the equation y”+$f(x)$y2n-1=0. Ca\check{}sopis pěst mat. 85, 357-358 (1960) · Zbl 0129.06204 [8] J.S.W. Wong, A nonoscillation theorem for sublinear Emden--Fowler equation, Anal. Appl., to appear · Zbl 1051.34028 [9] Wong, J. S. W.: A nonoscillation theorem for Emden--Fowler equations. J. math. Anal. appl. (2003) · Zbl 1051.34028