Nonoscillation theorems for second order nonlinear differential equations
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- by James S. W. Wong PDF
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Abstract:
We prove nonoscillation theorems for the second order Emden-Fowler equation (E): $y''+a(x)|y|^{\gamma -1}y=0$, $\gamma >0$, where $a(x)\in C(0,\infty )$ and $\gamma \not =1$. It is shown that when $x^{(\gamma +3)/2+\delta }a(x)$ is nondecreasing for any $\delta >0$ and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when $0<\gamma <1$ and $0<\delta <(1-\gamma )/2$.References
- S. Belohorec, Oscillatory solutions of certain nonlinear differential equations of second order, Mat. Fyz. Casopis Solven. Akad. Vied. 11 (1961), 250–255. (in Czech).
- Štefan Belohorec, On some properties of the equation $y^{\prime \prime }(x)+f(x)y^{\alpha }(x)=0$, $0<\alpha <1$, Mat. Časopis Sloven. Akad. Vied 17 (1967), 10–19 (English, with Russian summary). MR 214854
- Kuo Liang Chiou, The existence of oscillatory solutions for the equation $d^{2}y/dt^{2}+q(t)y^{r}=0,\,0<r<1$, Proc. Amer. Math. Soc. 35 (1972), 120–122. MR 301292, DOI 10.1090/S0002-9939-1972-0301292-2
- 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. MR 296413, DOI 10.1090/S0002-9947-1972-0296413-9
- C. V. Coffman and J. S. W. Wong, Oscillation and nonoscillation theorems for second order ordinary differential equations, Funkcial. Ekvac. 15 (1972), 119–130. MR 333337
- Lynn H. Erbe and James S. Muldowney, On the existence of oscillatory solutions to nonlinear differential equations, Ann. Mat. Pura Appl. (4) 109 (1976), 23–38. MR 481254, DOI 10.1007/BF02416952
- Lynn H. Erbe and James S. Muldowney, Nonoscillation results for second order nonlinear differential equations, Rocky Mountain J. Math. 12 (1982), no. 4, 635–642. MR 683858, DOI 10.1216/RMJ-1982-12-4-635
- H. E. Gollwitzer, Nonoscillation theorems for a nonlinear differential equation, Proc. Amer. Math. Soc. 26 (1970), 78–84. MR 259243, DOI 10.1090/S0002-9939-1970-0259243-3
- J. W. Heidel, Uniqueness, continuation, and nonoscillation for a second order nonlinear differential equation, Pacific J. Math. 32 (1970), 715–721. MR 259244, DOI 10.2140/pjm.1970.32.715
- J. W. Heidel and Don B. Hinton, The existence of oscillatory solutions for a nonlinear differential equation, SIAM J. Math. Anal. 3 (1972), 344–351. MR 340721, DOI 10.1137/0503032
- Miloš Jasný, On the existence of an oscillating solution of the nonlinear differential equation of the second order $y^{\prime \prime }+f(x)y^{2n-1}=0,$ $f(x)>0$, Časopis Pěst. Mat. 85 (1960), 78–83 (Russian, with Czech and English summaries). MR 0142840, DOI 10.21136/CPM.1960.108129
- I. T. Kiguradze, A note on the oscillation of solutions of the equation $u^{\prime \prime }+a(t)| u| ^{n}\,\textrm {sgn}\,u=0$, Časopis Pěst. Mat. 92 (1967), 343–350 (Russian, with Czech and German summaries). MR 0221012
- I. T. Kiguradze, On the oscillatory and monotone solutions of ordinary differential equations, Arch. Math. (Brno) 14 (1978), no. 1, 21–44. MR 512742
- Jaroslav Kurzweil, A note on oscillatory solution of equation $y''+f(x)y^{2n-1}=0$, Časopis Pěst. Mat. 85 (1960), 357–358 (Russian, with English and Czech summaries). MR 0126025, DOI 10.21136/CPM.1960.117339
- Man Kam Kwong and J. S. W. Wong, Nonoscillation theorems for a second order sublinear ordinary differential equation, Proc. Amer. Math. Soc. 87 (1983), no. 3, 467–474. MR 684641, DOI 10.1090/S0002-9939-1983-0684641-2
- S. I. Pohožaev, On the eigenfunctions of the equation $\Delta u+\lambda f(u)=0$, Dokl. Akad. Nauk SSSR 165 (1965), 36–39 (Russian). MR 0192184
- James S. W. Wong, On the generalized Emden-Fowler equation, SIAM Rev. 17 (1975), 339–360. MR 367368, DOI 10.1137/1017036
- James S. W. Wong, Remarks on nonoscillation theorems for a second order nonlinear differential equation, Proc. Amer. Math. Soc. 83 (1981), no. 3, 541–546. MR 627687, DOI 10.1090/S0002-9939-1981-0627687-0
Additional Information
- James S. W. Wong
- Affiliation: Chinney Investments Ltd., Hong Kong; City University of Hong Kong, Hong Kong
- Received by editor(s): August 7, 1997
- Published electronically: January 28, 1999
- Communicated by: Hal L. Smith
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 1387-1395
- MSC (1991): Primary 34C10, 34C15
- DOI: https://doi.org/10.1090/S0002-9939-99-05036-4
- MathSciNet review: 1622997