Nonoscillation theorems for second order

nonlinear differential equations

Author:
James S. W. Wong

Journal:
Proc. Amer. Math. Soc. **127** (1999), 1387-1395

MSC (1991):
Primary 34C10, 34C15

Published electronically:
January 28, 1999

MathSciNet review:
1622997

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove nonoscillation theorems for the second order Emden-Fowler equation (E): , , where and . It is shown that when is nondecreasing for any and is bounded above, then (E) is nonoscillatory. This improves a well-known result of Belohorec in the sublinear case, i.e. when and .

**1.**S. Belohorec,*Oscillatory solutions of certain nonlinear differential equations of second order*, Mat. Fyz. Casopis Solven. Akad. Vied.**11**(1961), 250-255. (in Czech).**2.**Štefan Belohorec,*On some properties of the equation 𝑦′′(𝑥)+𝑓(𝑥)𝑦^{𝛼}(𝑥)=0, 0<𝛼<1*, Mat. Casopis Sloven. Akad. Vied**17**(1967), 10–19 (English, with Russian summary). MR**0214854****3.**Kuo Liang Chiou,*The existence of oscillatory solutions for the equation 𝑑²𝑦/𝑑𝑡²+𝑞(𝑡)𝑦^{𝑟}=0,0<𝑟<1*, Proc. Amer. Math. Soc.**35**(1972), 120–122. MR**0301292**, 10.1090/S0002-9939-1972-0301292-2**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. MR**0296413**, 10.1090/S0002-9947-1972-0296413-9**5.**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**0333337****6.**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**0481254****7.**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**, 10.1216/RMJ-1982-12-4-635**8.**H. E. Gollwitzer,*Nonoscillation theorems for a nonlinear differential equation*, Proc. Amer. Math. Soc.**26**(1970), 78–84. MR**0259243**, 10.1090/S0002-9939-1970-0259243-3**9.**J. W. Heidel,*Uniqueness, continuation, and nonoscillation for a second order nonlinear differential equation*, Pacific J. Math.**32**(1970), 715–721. MR**0259244****10.**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**0340721****11.**Miloš Jasný,*On the existence of an oscillating solution of the nonlinear differential equation of the second order 𝑦′′+𝑓(𝑥)𝑦²ⁿ⁻¹=0, 𝑓(𝑥)>0*, Časopis Pěst. Mat.**85**(1960), 78–83 (Russian, with Czech and English summaries). MR**0142840****12.**I. T. Kiguradze,*A note on the oscillation of solutions of the equation 𝑢′′+𝑎(𝑡)\vert𝑢\vertⁿ𝑠𝑔𝑛𝑢=0*, Časopis Pěst. Mat.**92**(1967), 343–350 (Russian, with Czech and German summaries). MR**0221012****13.**I. T. Kiguradze,*On the oscillatory and monotone solutions of ordinary differential equations*, Arch. Math. (Brno)**14**(1978), no. 1, 21–44. MR**512742****14.**Jaroslav Kurzweil,*A note on oscillatory solution of equation 𝑦”+𝑓(𝑥)𝑦²ⁿ⁻¹=0*, Časopis Pěst. Mat.**85**(1960), 357–358 (Russian, with English and Czech summaries). MR**0126025****15.**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**, 10.1090/S0002-9939-1983-0684641-2**16.**S. I. Pohožaev,*On the eigenfunctions of the equation Δ𝑢+𝜆𝑓(𝑢)=0*, Dokl. Akad. Nauk SSSR**165**(1965), 36–39 (Russian). MR**0192184****17.**James S. W. Wong,*On the generalized Emden-Fowler equation*, SIAM Rev.**17**(1975), 339–360. MR**0367368****18.**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**, 10.1090/S0002-9939-1981-0627687-0

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Additional Information

**James S. W. Wong**

Affiliation:
Chinney Investments Ltd., Hong Kong;
City University of Hong Kong, Hong Kong

DOI:
http://dx.doi.org/10.1090/S0002-9939-99-05036-4

Keywords:
Second order,
nonlinear,
ordinary differential equations,
oscillation,
asymptotic behavior

Received by editor(s):
August 7, 1997

Published electronically:
January 28, 1999

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 1999
American Mathematical Society