INTERVAL CRITERIA FOR FORCED OSCILLATION OF DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN AND NONLINEARITIES GIVEN BY RIEMANN-STIELTJES INTEGRALS

- Journal title : Journal of the Korean Mathematical Society
- Volume 49, Issue 5, 2012, pp.1017-1030
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2012.49.5.1017

Title & Authors

INTERVAL CRITERIA FOR FORCED OSCILLATION OF DIFFERENTIAL EQUATIONS WITH p-LAPLACIAN AND NONLINEARITIES GIVEN BY RIEMANN-STIELTJES INTEGRALS

Hassan, Taher S.; Kong, Qingkai;

Hassan, Taher S.; Kong, Qingkai;

Abstract

We consider forced second order differential equation with -Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of , where , , , is strictly increasing such that <<, , , with > on , , and is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature.

Keywords

interval criteria;forced oscillation;-Laplacian;nonlinear differential equations;

Language

English

Cited by

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References

1.

R. P. Agarwal, S. R. Grace, and D. O'Regan, Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations, Kluwer Academic, Dordrecht, 2002.

2.

E. F. Beckenbach and R. Bellman, Inequalities, Springer, Berlin, 1961.

3.

G. J. Butler, Oscillation theorems for a nonlinear analogue of Hill's equation, Quart. J. Math. Oxford Ser. (2) 27 (1976), no. 106, 159-171.

4.

G. J. Butler, Integral averages and the oscillation of second order ordinary differential equations, SIAM J. Math. Anal. 11 (1980), no. 1, 190-200.

5.

D. Cakmak and A. Tiryaki, Oscillation criteria for certain forced second order nonlinear differential equations with delayed argument, Comput. Math. Appl. 49 (2005), no. 11-12, 1647-1653.

6.

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.

7.

E. M. Elabbasy and T. S. Hassan, Interval oscillation for second order sublinear differ- ential equations with a damping term, Int. J. Dyn. Syst. Differ. Equ. 1 (2008), no. 4, 291-299.

8.

E. M. Elabbasy, T. S. Hassan, and S. H. Saker, Oscillation of second-order nonlinear differential equations with a damping term, Electron. J. Differential Equations 2005 (2005), No. 76, 13 pp.

9.

M. A. El-Sayed, An oscillation criterion for a forced second order linear differential equation, Proc. Amer. Math. Soc. 118 (1993), no. 3, 813-817.

10.

L. Erbe, T. S. Hassan, and A. Peterson, Oscillation of second order neutral delay differential equations, Adv. Dyn. Syst. Appl. 3 (2008), no. 1, 53-71.

11.

A. F. Guvenilir and A. Zafer, Second order oscillation of forced functional differential equations with oscillatory potentials, Comput. Math. Appl. 51 (2006), no. 9-10, 1395-1404.

12.

G. H. Hardy, J. E. Littlewood, and G. Polya, Inequalities, Second ed., Cambridge University Press, Cambridge, 1988.

13.

T. S. Hassan, Interval oscillation for second order nonlinear differential equations with a damping term, Serdica Math. J. 34 (2008), no. 4, 715-732.

14.

T. S. Hassan, L. Erbe, and A. Peterson, Forced oscillation of second order functional differential equations with mixed nonlinearities, Acta Mathematica Scientia 31B (2011), no. 2, 613-626.

15.

T. S. Hassan and Q. Kong, Interval criteria for forced oscillation of differential equations with p-Laplacian, damping, and mixed nonlinearities, Dynamic Systems & Applications 20 (2011), 279-294.

16.

A. G. Kartsatos, On the maintenance of oscillations of nth order equations under the effect of a small forcing term, J. Differential Equations 10 (1971), 355-363.

17.

A. G. Kartsatos, Maintenance of oscillations under the effect of a periodic forcing term, Proc. Amer. Math. Soc. 33 (1972), 377-383.

18.

M. S. Keener, On the solutions of certain linear nonhomogeneous second-order differ- ential equations, Applicable Anal. 1 (1971), no. 1, 57-63.

19.

Q. Kong, Interval criteria for oscillation of second-order linear ordinary differential equations, J. Math. Anal. Appl. 229 (1999), no. 1, 258-270.

20.

Q. Kong, Oscillation criteria for second order half-linear differential equations, Differential equations with applications to biology (Halifax, NS, 1997), 317-323, Fields Inst. Commun., 21, Amer. Math. Soc., Providence, RI, 1999.

21.

Q. Kong and J. S. W. Wong, Oscillation of a forced second order differential equations with a deviating argument, Funct. Differ. Equ. 17 (2010), no. 1-2, 141-155.

22.

Q. Kong and B. G. Zhang, Oscillation of a forced second order nonlinear equation, Chinese Ann. Math. Ser. B 15 (1994), no. 1, 59-68.

23.

M. K. Kwong and J. S. W. Wong, Linearization of second order nonlinear oscillation theorems, Trans. Amer. Math. Soc. 279 (1983), no. 2, 705-722.

24.

A. H. Nasr, Sufficient conditions for the oscillation of forced super-linear second order differential equations with oscillatory potential, Proc. Amer. Math. Soc. 126 (1998), no. 1, 123-125.

25.

C. H. Ou and J. S. W. Wong, Forced oscillation of nth-order functional differential equations, J. Math. Anal. Appl. 262 (2001), no. 2, 722-731.

26.

Ch. G. Philos, Oscillation theorems for linear differential equations of second order, Arch. Math. (Basel) 53 (1989), no. 5, 482-492.

27.

S. M. Rankin, Oscillation theorems for second order nonhomogeneous linear differential equations, J. Math. Anal. Appl. 53 (1976), no. 3, 550-553.

28.

A. Skidmore and J. J. Bowers, Oscillatory behavior of solutions of y′' + p(x)y = f(x), J. Math. Anal. Appl. 49 (1975), 317-323.

29.

A. Skidmore and W. Leighton, On the differential equation y"+p(x)y = f(x), J. Math. Anal. Appl. 43 (1973), 46-55.

31.

Y. G. Sun and Q. Kong, Interval criteria for forced oscillation with nonlinearities given by Riemann-Stieltjes integrals, Comput. Math. Appl. 62 (2011), no. 1, 243-252.

32.

Y. G. Sun and F. W. Meng, Interval criteria for oscillation of second order differential equations with mixed nonlinearities, Appl. Math. Comp. 198 (2008), no. 1, 375-381.

33.

Y. G. Sun, C. H. Ou, and J. S. W. Wong, Interval oscillation theorems for a linear second-order differential equation, Comput. Math. Appl. 48 (2004), no. 10-11, 1693-1699.

34.

Y. G. Sun and J. S. W. Wong, Note on forced oscillation of nth-order sublinear differ- ential equations, J. Math. Anal. Appl. 298 (2004), no. 1, 114-119.

35.

Y. G. Sun and J. S. W. Wong, Oscillation criteria for second order forced ordinary differential equations with mixed nonlinearities, J. Math. Anal. Appl. 334 (2007), no. 1, 549-560.

37.

J. S. W. Wong, Second order nonlinear forced oscillations, SIAM J. Math. Anal. 19 (1988), no. 3, 667-675.

38.

J. S. W. Wong, Oscillation criteria for a forced second-order linear differential equation, J. Math. Anal. Appl. 231 (1999), no. 1, 235-240.

39.

Q. Yang, Interval oscillation criteria for a forced second order nonlinear ordinary differential equations with oscillatory potential, Appl. Math. Comput. 136 (2003), no. 1, 49-64.