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Keywords:
second order nonlinear differential equation; oscillatory solution; nonoscillatory solution; coexistence problem
second order nonlinear differential equation; oscillatory solution; nonoscillatory solution; coexistence problem
Summary:
Oscillatory properties of the second order nonlinear equation \[ (r(t)x^{\prime })^{\prime }+q(t)f(x)=0 \] are investigated. In particular, criteria for the existence of at least one oscillatory solution and for the global monotonicity properties of nonoscillatory solutions are established. The possible coexistence of oscillatory and nonoscillatory solutions is studied too.
References:
[1] M. Cecchi, M. Marini and G. Villari: On some classes of continuable solutions of a nonlinear differential equation. J. Diff. Equat. 118 (1995), 403–419. DOI 10.1006/jdeq.1995.1079 | MR 1330834
[2] C. V. Coffman and J. S. W. Wong: On a second order nonlinear oscillation problem. Trans. Amer. Math. Soc. 147 (1970), 357–366. DOI 10.1090/S0002-9947-1970-0257473-2 | MR 0257473
[3] C. V. Coffman and J. S. W. Wong: Oscillation and nonoscillation theorems for second order differential equations. Funkcialaj Ekvacioj 15 (1972), 119–130. MR 0333337
[4] C. V. Coffman and J. S. W. Wong: Oscillation and nonoscillation of solutions of generalized Emden-Fowler equations. Trans. Amer. Math. Soc. 167 (1972), 399–434. DOI 10.1090/S0002-9947-1972-0296413-9 | MR 0296413
[5] L. H. Erbe and J. S. Muldowney: On the existence of oscillatory solutions to nonlinear differential equations. Ann. Mat. Pura Appl. 59 (1976), 23–37. MR 0481254
[6] L. H. Erbe and H. Lu: Nonoscillation theorems for second order nonlinear differential equations. Funkcialaj Ekvacioj 33 (1990), 227–244. MR 1078128
[7] L. H. Erbe: Nonoscillation criteria for second order nonlinear differential equations. J. Math. Anal. Appl. 108 (1985), 515–527. DOI 10.1016/0022-247X(85)90042-3 | MR 0793663 | Zbl 0579.34027
[8] J. W. Heidel and I. T. Kiguradze: Oscillatory solutions for a generalized sublinear second order differential equation. Proc. Amer. Math. Soc. 38 (1973), 80–82. DOI 10.1090/S0002-9939-1973-0310339-X | MR 0310339
[9] D. V. Izjumova: On oscillation and nonoscillation conditions for solutions of nonlinear second order differential equations. Diff. Urav. 11 (1966), 1572–1586. (Russian)
[10] M. Jasný: On the existence of oscillatory solutions of the second order nonlinear differential equation $y^{\prime \prime }+f(x)y^{2n-1}=0$. Čas. Pěst. Mat. 85 (1960), 78–83. (Russian)
[11] I. Kiguradze and A. Chanturia: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Acad. Publ., , 1993. MR 1220223
[12] J. Kurzweil: A note on oscillatory solutions of the equation $y^{\prime \prime }+f(x)y^{2n-1}=0$. Čas. Pěst. Mat. 82 (1957), 218–226. (Russian)
[13] Chiou Kuo-Liang: The existence of oscillatory solutions for the equation ${\mathrm d}^2y/{\mathrm d}t^2 +q(t)y^{r}=0$, $0. Proc. Amer. Math. Soc. 35 (1972), 120–122. DOI 10.1090/S0002-9939-1972-0301292-2 | MR 0301292
[14] S. Matucci: On asymptotic decaying solutions for a class of second order differential equations. Arch. Math. (Brno) 35 (1999), 275–284. MR 1725843 | Zbl 1048.34088
[15] J. D. Mirzov: Asymptotic properties of solutions of the systems of nonlinear nonautonomous ordinary differential equations. (1993), Adygeja Publ., Maikop. (Russian)
[16] R. A. Moore and Z. Nehari: Nonoscillation theorems for a class of nonlinear differential equations. Trans. Amer. Math. Soc. 93 (1959), 30–52. DOI 10.1090/S0002-9947-1959-0111897-8 | MR 0111897
[17] J. Sugie and K. Kita: Oscillation criteria for second order nonlinear differential equations of Euler type. J. Math. Anal. Appl. 253 (2001), 414–439. DOI 10.1006/jmaa.2000.7149 | MR 1808146
[18] J. S. W. Wong: On the generalized Emden-Fowler equation. SIAM Rev. 17 (1975), 339–360. DOI 10.1137/1017036 | MR 0367368 | Zbl 0295.34026
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