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

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

Published electronically:
January 28, 1999

MathSciNet review:
1622997

Full-text PDF

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.**S. Belohorec,*On some properties of the equation ,*, ibid.**17**(1967), 10-19. MR**35:5703****3.**K. L. Chiou,*The existence of oscillatory solutions for the equation ,*, Proc. Amer. Math. Soc.**35**(1972), 120-122. MR**46:450****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**45:5473****5.**C. V. Coffman and J. S. W. Wong,*Oscillation and nonoscillation theorems for second order differential equations*, Funkcial. Ekvac.**15**(1972), 119-130. MR**48:11662****6.**L. H. Erbe and J. S. Muldowney,*On the existence of oscillatory solutions to nonlinear differential equations*, Annali di Matematica Pure ed Applicata**109**(1976), 23-38. MR**58:1380****7.**L. H. Erbe and J. S. Muldowney,*Nonoscillation results for second order nonlinear differential equation*, Rocky Mountain Math. J.**12**(1982), 635-642. MR**84a:34031****8.**H. E. Gollwitzer,*Nonoscillation theorems for a nonlinear differential equation*, Proc. Amer. Math. Soc.**26**(1970), 78-84. MR**41:3885****9.**J. W. Heidel,*Uniqueness, continuous and nonoscillation for a second order nonlinear differential equation*, Pacific J. Math.**32**(1970), 715-721. MR**41:3886****10.**J. W. Heidel and D. B. Hinton,*The existence of oscillatory solutions for a nonlinear differential equation*, SIAM J. Math. Anal.**3**(1972), 344-351. MR**49:5472****11.**M. Jasny,*On the existence of an oscillatory solution of the nonlinear differential equation of the second order ,*, Casopis Pest Mat.**85**(1960), 78-83. MR**26:408****12.**I. T. Kiguradze,*A note on the oscillation of solution of the equation*, Casopis Pest Mat.**92**(1967), 343-350. (in Russian). MR**36:4064****13.**I. T. Kiguradze,*On the oscillatory and monotone solutions of ordinary differential equations*, Arch. Math. Scripta Fac. Sci. Nat.**14**(1978), 21-44. MR**80b:34031****14.**J. Kurzweil,*A note on oscillatory solutions of the equations*, Casopis Pest Mat.**85**(1960), 357-358. (Russian). MR**23:A3322****15.**M. K. Kwong and J. S. W. Wong,*Nonoscillation theorems for a second order sublinear ordinary differential equation*, Proc. Amer. Math. Soc.**87**(1983), 467-474. MR**84b:34039****16.**S. I. Pohozaev,*Eigenfunctions of the equation*, Dokl. Akad. Nauk SSSR**165**(1965), 36-39 (in Russian) and Soviet Math.**6**(1965), 1408-1411 (in English). MR**33:411****17.**J. S. W. Wong,*On the generalized Emden-Fowler equation*, SIAM Review**17**(1975), 339-360. MR**51:3610****18.**J. S. W. Wong,*Remarks on nonoscillation theorems for a second order nonlinear differential equation*, Proc. Amer. Math. Soc.**83**(1981), 541-546. MR**82i:34034**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34C10,
34C15

Retrieve articles in all journals with MSC (1991): 34C10, 34C15

Additional Information

**James S. W. Wong**

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

DOI:
https://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