Oscillation Theorems for Nonlinear Second Order Forced Differential Equations
Keywords:
Oscillation, Forced Nonlinear differential equations of second orderAbstract
Relating to the oscillation theory, in the present paper, we consider a class of forced nonlinear differential equations of second order. However, we discuss the problem of finding sufficient criteria for all solutions of these equations to be oscillate. By employing a generalized Ricati technique and also using an integral averaging technique, we derive several new oscillation theorems. Our results obtained here generalize and improve some of well-known ones in the literature. Some carefully selected examples are also given to illustrate the effect of impulses on the oscillatory behavior of all solutions for this class.
References
R.P, Agarwal, C. Avramescu, O.G. Mustafa, On the oscillation theory of a second-order strictly sublinear differential equation. Can. Math. Bull. 53(2), 193-203, 2010.
Xh. Beqiri, E. Koci, oscillation criteria for second order nonlinear differential equations, British Journal of Science, Vol. 6(2). pg. 73-80, 2012.
I. Bihari, An oscillation theorem concerning the half linear differential equation of the Second order, Magyar Tud. Akad. Mat. Kutato Int. Kozl. 8, 275-280, 1963.
E. M. Elabbasy,M. A Elsharabasy, Oscillation properties for second order nonlinear differential equations, Kyungpook Math. J. 37 211-220, 1997.
E. M. Elabbasy,W.W.Elhaddad, Oscillation of second order nonlinear differential equations with damping term. Electronic Journal of Qualitative Theory of Differential Equations. 25, 1-19, 2007.
S. R. Grace, B. S. Lalli and C. C. Yeh, Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term, SIAM J. Math. Anal., 15, 1082- 1093,1984.
S.R.Grace, B.S.Lalli, On the second order nonlinear oscillations, Bull. Inst. Math. Acad. Sinica 15, no. 3, 297-309, 1987.
S. R. Grace, B. S. Lalli and C. C. Yeh, Addendum: Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term, SIAM J. Math. Anal.19, no. 5, 1252-1253, 1988.
S. R. Grace, Oscillation theorems for second order nonlinear differential equations wit damping, math. Nachr. 141, 117-127, 1989.
S.R.Grace, B.S.Lalli, Oscillation theorems for second order nonlinear differential equations with a damping term, Comment. Math. Univ. Carolinae 30, no. 4, 691- 697, 1989.
S. R. Grace, Oscillation criteria for second order differential equations with damping J. Austral. Math. Soc. (Series A) 49, 43-54, 1990.
S.R.Grace, B.S.Lalli, Integral averaging technique for the oscillation of second order nonlinear differential equations, J. Math. Anal. Appl. 149, 277-311, 1990.
S.R. Grace, Oscillation theorems for nonlinear differential equations of second order, Math. Anal. And Appl. 171, 220-241, 1992.
J. R. Greaf, P. W. Spikes,” On the oscillatory behaviour of solutions of second order nonlinear differential equation, “ Czech. Math. J. 36, 275-284, 1986.
Graef, JR, Rankin, SM, Spikes, PW: Oscillation theorems for perturbed non-linear differential equation. J. Math. Anal. Appl. 65, 375-390, 1978.
C. F. Lee, C. C. Yeh, An Oscillation theorems. Applied Mathematics Letters. 20, 238- 240, 2007.
I.V. Kamenev, Integral criterion for oscillation of linear differential equations of second order, Math. Zametki. 23, 249-251, 1978.
A. G. Kartsatos, On oscillation of nonlinear equations of second order, J. Math. Anal. Appl. 24, 665- 668, 1968.
W.T. Li, R.P. Agarwal, Interval oscillation criteria for second order nonlinear equations with damping, Computers Math. Applic. 40, 217-230, 2000.
F.W. Meng, An oscillation theorem for second order superlinear differential equations, Ind. J. Pure Appl. Math. 27, 651–658, 1996.
Y. Nagabuchi, M. Yamamoto, Some oscillation criteria for second order nonlinear ordinary differential equations with damping, Proc. Japan Acad. 64, 282-285, 1988.
J.Ohriska , Antonia Zulova, Oscillation criteria for second order nonlinear differential equation, IM Preprint series A, Vol. 10, 1-11, 2004.
Z, Ouyang, J, Zhong, S, Zou, Oscillation criteria for a class of second-order nonlinear differential equations with damping term. Abstr. Appl. Anal. 2009, 1-12, 2009.
Ch. G. Philos, Oscillation of sublinear differential equations of second order, Nonlinear Anal. 7, no. 10, 1071-1080, 1983.
Ch. G. Philos, On second order sublinear oscillation, Aequations Math. 27, 242- 254, 1984.
Ch. G. Philos, Integral averages and second order superlinear oscillation, Math, Nachr. 120, 127-138, 1985.
M. Remili, “Oscillation criteria for second order nonlinear perturbed differential equations,” Electronic Journal of Qualitative Theory of Differential Equations.25,1-11, 2010.
A. A. Salhin, Oscillation criteria of second order nonlinear differential equations with variable Coefficients, Discrete Dynamics in Nature and Society. 2014 (2014), 1-9.
A, Tiryaki, Y, Basci, Oscillation theorems for certain even-order nonlinear damped differential equations. Rocky Mt. J. Math. 38(3), 1011-1035, 2008.
A, Tiryaki, Oscillation criteria for a certain second-order nonlinear differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ. 61, 1-11, 2009.
P. Temtek and A. Tiryaki, “Oscillation criteria for a certain second-order nonlinear perturbed differential equations,” Journal of Inequalities and Applications 524,1-12, 2013.
J. Yan, Oscillation theorems for second order linear differential equations with damping, Proc. Amer. Math. Soc. 98, no. 2, 276-282, 1986.
S. Yibing, H. Zhenlai, S. Shurong, and Z. Chao, Interval Oscillation Criteria for Second-Order Nonlinear Forced Dynamic Equations with Damping on Time Scales, Abstr. Appl. Anal., 2013, 1-11, 2013.
S. Yibing, H. Zhenlai, S. Shurong, and Z. Chao,, Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping, Abstr. Appl. Anal., 2013, 1-10, 2012.
Q, Zhang, L, Wang, Oscillatory behavior of solutions for a class of second-order nonlinear differential equation with perturbation. Acta Appl. Math. 110, 885-893, 2010.
