Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses
American Journal of Applied Mathematics
Volume 5, Issue 4, August 2017, Pages: 119-123
Received: Oct. 20, 2016;
Accepted: Nov. 14, 2016;
Published: Aug. 24, 2017
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Ubon Akpan Abasiekwere, Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria
Edwin Frank Nsien, Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria
Imoh Udo Moffat, Department of Mathematics and Statistics, Faculty of Science, University of Uyo, Uyo, Nigeria
In this paper, we study a certain type of second order linear neutral differential equation with constant impulsive jumps. This type of equation is known always to possess an unbounded non-oscillatory solution. The method and technique of impulse imposition used here is due to studies by Bainov and Simeonov . By assuming, amongst other conditions, that the constant coefficient of the equation in question lies between zero and one and the delay function is non-decreasing, it is shown that all bounded solutions of the said neutral impulsive equation are oscillatory.
Ubon Akpan Abasiekwere,
Edwin Frank Nsien,
Imoh Udo Moffat,
Oscillation Conditions for a Type of Second Order Neutral Differential Equations with Impulses, American Journal of Applied Mathematics.
Vol. 5, No. 4,
2017, pp. 119-123.
D. D. Bainov and P. S. Simeonov, Oscillation Theory of Impulsive Differential Equations, International Publications Orlando, Florida, 1998.
R. D. Driver, A mixed neutral system, Nonlinear Anal. 8, 1984, pp. 155–158.
G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987.
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
W. T. Li, Interval oscillation criteria of second-order half-linear functional differential equations, Appl. Math. Comput. 155, 2004, pp. 451–468.
X. Lin, Oscillation of second-order nonlinear neutral differential equations, J. Math. Anal. Appl. 309, 2005, pp. 442–452.
F. Meng and J. Wang, Oscillation criteria for second order quasilinear neutral delay differential equations, J. Indones. Math. Soc. (MIHMI) 10, 2004, pp. 61–75.
I. O. Isaac and Z. Lipcsey, Oscillations of Scalar Neutral Impulsive Differential Equations of the First Order with variable Coefficients, Dynamic Systems and Applications, 19, 2010, pp. 45-62.
I. O. Isaac, Z. Lipcsey and U. J. Ibok, Nonoscillatory and Oscillatory Criteria for First Order Nonlinear Neutral Impulsive Differential Equations, Journal of Mathematics Research; Vol. 3 Issue 2, 2011, pp. 52-65.
L. H. Erbe, Q. Kong and B. G. Zhang, Oscillation Theory for Functional Differential Equations, Dekker, New York, 1995.
I. O. Isaac and Z. Lipcsey, Oscillations in Linear Neutral Delay Impulsive Differential Equations with Constant Coefficients, Communications in Applied Analysis, 14, 2010, pp. 123-136.
I. O. Isaac and Z. Lipcsey, Linearized Oscillations in Nonlinear Neutral Delay Impulsive Differential Equations, Journal of Modern Mathematics and Statistics, 3, 2009, pp. 17-21.
I. O. Isaac and Z. Lipcsey, Oscillatory Conditions on Both Directions for a Nonlinear Impulsive Diﬀerential Equation with Deviating Arguments, Journal of Mathematics Research, 3, 2011, pp. 48-51.
D. D. Bainov, M. B. Dimitrova and A. B. Dishliev, Oscillationof the Bounded Solutions of Impulsive Differential-Difference Equations of Second Order, Applied Mathematics and Computation, 114, 2000, pp. 61-68.
D. D. Bainov and P. S. Simeonov, The Second Method of Lyapunov for Systems with an Impulse Effect, Tamkang J. Math. 16, 1985, pp. 19-40.
D. D. Bainov and P. S. Simeonov, Stability with respect to Part of the Variables in Systema with Impulsive Effect, Journal of Mathematical Analysis and Applications, 1986, pp. 247-263.
C. Yong-Shao and F. Wei-Zhen, Oscillation of Second Order Nonlinear Ordinary Differential Equations with Impulses, Journal of Mathematical Analysis and Applications, 210, 1997, pp. 150-169.
A. B. Dishliev and D. D. Bainov, Dependence upon Initial Conditions and Parameter of Impulsive Differential Equations with Variable Structure, International Journal of Theoretical Physics, 29 No. 6, 1990, pp. 655-675.
S. I. Gurgula, Investigation of the Stability of Solutions of Impulse Systems by Lyapunov's Second Method, Ukrainian Math., 1, 1982, pp. 400-103.
S. V. Krishna, J. Vasundlara Devi and K. Satyavani, Boundedness and Dichotomies for Impulse Equations, Journal of Mathematical Analysis and Applications, 158, 1991, pp. 352-375.
G. K. Kulev and D. D. Bainov, Lipschitz Stability of Impulsive Systems of Differential Equations, International Journal of Theoretical Physics 30, 1991, pp. 737-756.
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Publishing Company Ltd, Singapore, 1989.
V. Lakshmikantham and X. Liu, On Quasistability for Impulsive Differential Systems, Nonlinear Analysis, 13, No. 7, 1989, pp. 819-828.
M. Peng and Ge, Oscillation Criteria for Second Order Nonlinear Differential Equations with Impulses, Computers and Mathematics with Applications, 30, 2000, pp. 217-225.
A. M. Samoilenko and N. A. Perestyuk, Stability of the Solutions of Differential Equations with Impulse Effect, Differential Equations, 11, 1977, pp. 1981-1992.
P. P. Zabreiko, D. D. Bainov and S. I. Kostadinov, Characteristic Exponents of Impulsive Differential Equations in a Banach Space, International Journal of Theoretical Physics, 27, 1988, pp. 721-743.
Y. Zhang, A. Zhao and J. Yan, Oscillation Criteria for Impulsive Differential Equations, Journal of Mathematical Analysis and Applications, 205, 1997, pp. 461-470.
I. O. Isaac, Z. Lipcsey and U. J. Ibok, Linearized Oscillations in Autonomous Delay Impulsive Differential Equations, British Journal of Mathematics & Computer Science, 4(21): 2014, pp. 3068-3076.
A. M. Samoilenko and N. A. Perestyuk, Impulsive differential Equations, World Scientific Publishing Company Ltd, Singapore, 1995.
R. P. Agarwal, M. Benchohra, D. O'Regan and A. Ouahab, Second order Impulsive dynamic equations on time scales, Functional differential equations II, No. 3-4, 2004, pp. 223–234.
S. G. Deo and S. G. Pandit, Differential systems involving impulses, Lecture Notes 954, Springer-Verlag, Berlin. differential equations Arch. Mat. 41, 1982, pp. 352-362.
J. Cheng and Y. Chu, Oscillations of Second-Order Neutral Impulsive Differential Equations, Journal of Inequalities and Applications; doi: 10.1155/2010/493927: 1-29.
E. M. Bonotto, L. P. Gimenes and M. Federson, Oscillation for a second-order neutral diﬀerential equation with impulses, Cadernos De Matematica; 09: 2008, pp. 169–190.
Y. Sun, Oscillation of Certain Second-Order Sub-Half-Linear Neutral Impulsive Differential Equations, Discrete Dynamics in Nature and Society; doi: 10.1155/2011/195619: 2011, pp. 1-10.
A. Tripathy and S. Santra, Oscillation Properties of a Class of Second Order Impulsive Differential Systems of Neutral Type, Functional Differential Equations; 23(1-2): 2016, pp. 57–71.
D. D. Bainov and M. B. Dimitrova, Oscillation of sub and super linear impulsive diﬀerential equations with constant delay, Applicable Analysis; 64, 1977, pp. 57–67.
U. A. Abasiekwere and I. U. Moffat, Oscillation Theorems for Linear Neutral Impulsive Differential Equations of the Second Order with Variable Coefficients and Constant Retarded Arguments, Applied Mathematics, Vol. 7 No. 3, 2017, pp. 39-43. doi: 10.5923/j.am.20170703.01.
U. A. Abasiekwere, E. F. Nsien and I. U. Moffat, On the Existence of Bounded Oscillatory Solutions of Impulsive Delay Differential Equations of the Second Order, International Journal of Mathematics Trends and Technology, 2017, 48(2), pp. 6-10, ISSN: 2231-5373.