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

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;

Abstract
We consider forced second order differential equation with $\small{p}$-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of $$(p(t){\phi}_{\gamma}(x^{\prime}(t)))^{\prime}+q_0(t){\phi}_{\gamma}(x(t))+{\int}^b_0q(t,s){\phi}_{{\alpha}(s)}(x(t))d{\zeta}(s) Keywords interval criteria;forced oscillation;$\small{p}$-Laplacian;nonlinear differential equations; Language English Cited by 1. Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals, Advances in Difference Equations, 2012, 2012, 1, 175 2. Oscillation Criteria for Functional Nonlinear Dynamic Equations with$${\gamma} γ -Laplacian, Damping and Nonlinearities Given by Riemann–Stieltjes Integrals, Mediterranean Journal of Mathematics, 2016, 13, 3, 981
3.
Comparison criteria for odd order forced nonlinear functional neutral dynamic equations, Applied Mathematics and Computation, 2015, 251, 387
4.
Oscillation criteria for higher order nonlinear dynamic equations, Mathematische Nachrichten, 2014, 287, 14-15, 1659
5.
Comparison criteria for third order functional dynamic equations with mixed nonlinearities, Applied Mathematics and Computation, 2015, 268, 169
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.

30.
Y. G. Sun, A note on Nasr's and Wong's papers, J. Math. Anal. Appl. 286 (2003), no. 1, 363-367.

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.

36.
H. Teufel, Forced second order nonlinear oscillations, J. Math. Anal. Appl. 40 (1972), 148-152.

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.