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August  2017, 22(6): 2351-2363. doi: 10.3934/dcdsb.2017103

Oscillation theorems for impulsive parabolic differential system of neutral type

Min Zou 1, , An-Ping Liu 1, and  Zhimin Zhang 2,,

1. 

School of Mathematics and Physics, China University of Geosciences, Wuhan, Hubei 430074, China
2. 

Department of Mathematics, Wayne State University, Detroit, MI 48202, USA

E-mail address: zzhang@math.wayne.edu

Received  June 2016 Revised  July 2016 Published  March 2017

Fund Project: This work is supported in part by the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) No.2012129100, National Natural Science Foundation of China No.11201436, No.91430216, and No.11471031, and National Basic Research Program of China (973 Program)(2011CB710604).

In this paper, oscillatory properties of solutions to a nonlinear impulsive parabolic differential system of neutral type are investigated. A series of sufficient conditions are established for problems with Robin and Dirichlet boundary conditions. Examples are provided to confirm the validity of the analysis.

Keywords: Oscillation, parabolic differential system, impulsive differential equation, delay, neutral type, nonlinear problem.
Mathematics Subject Classification: Primary: 58J45, 35B05; Secondary: 35R1.
Citation: Min Zou, An-Ping Liu, Zhimin Zhang. Oscillation theorems for impulsive parabolic differential system of neutral type. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2351-2363. doi: 10.3934/dcdsb.2017103
References:
[1]

D. D. Bainov and E. Minchev, Oscillation of the solutions of impulsive parabolic equations, Journal of Computational and Applied Mathematics, 69 (1996), 207-214.

[2]

B. T. Cui and M. A. Han, Oscillation theorems for nonlinear hyperbolic systems with impulses, Nonlinear Analysis: Real World Applications, 9 (2008), 94-102.

[3]

B. T. Cui and Y. H. Yu, Oscillations of certain hyperbolic partial differential equations of neutral type, Acta Applicandae Mathematicae, 19 (1996), 80-88.

[4]

X. L. Fu and L. Q. Zhang, Force oscillation for impulsive hyperbolic boundary value problems with delay, Applied Mathematics and Computation, 158 (2004), 761-780.

[5]

M. X. He and A. P. Liu, Oscillation of hyperbolic functional differential equations, Applied Mathematics and Computation, 142 (2003), 205-224.

[6]

V. Lakshmikantham, D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[7]

W. N. Li, On the force oscillation of solutions for systems of impulsive parabolic differential equations with several delays, Journal of Computational and Applied Mathematics, 181 (2005), 46-57.

[8]

A. P. Liu, Oscillations of certain hyperbolic delay differential equations with damping term, Mathematics Applications, 9 (1996), 321-324.

[9]

A. P. Liu and S. C. Cao, Oscillations of the solutions of hyperbolic partial differential equations of neutral type, Chinese Quarterly Journal of Mathematics, 17 (2002), 7-13.

[10]

A. P. Liu and M. X. He, Oscillation of the solutions of nonlinear delay hyperbolic patrial differential equations of neutral type, Applied Mathematics and Mechanics, 23 (2002), 678-685.

[11]

A. P. Liu, T. Liu and M. Zou, Oscillation of nonlinear impulsive parabolic differential equations of neutral type, Rocky Mountain Journal of Mathematics, 41 (2011), 833-850.

[12]

A. P. Liu, Q. X. Ma and M. X. He, Oscillation of nonlinear impulsive parabolic equations of neutral type, Rocky Mountain Journal of Mathematics, 36 (2006), 1011-1026.

[13]

A. P. Liu, L. Xiao and T. Liu, Oscillation of nonlinear impulsive hyperbolic differential equations with several delays, Electronic Journal of Differential Equations, 24 (2004), 1-6.

[14]

J. R. Yan and C. H. Kou, Oscillation of solutions of delay impulsive differential equations, Journal of Mathematical Analysis and Applications, 254 (2001), 358-370.

[15]

M. Zou, T. Chang, Y. L. Li and A. P. Liu, Necessary and sufficient condition for oscillation of impulsive delay hyperbolic differential system, Annals of Differential Equations, 23 (2007), 608-611.

[16]

M. Zou, A. P. Liu, L. H. He and Y. L. Li, Oscillation for boundary value problem of impulsive delay parabolic differential system, Journal of Natural Science of Heilongjiang University, 23 (2006), 528-531.

show all references

References:
[1]

D. D. Bainov and E. Minchev, Oscillation of the solutions of impulsive parabolic equations, Journal of Computational and Applied Mathematics, 69 (1996), 207-214.

[2]

B. T. Cui and M. A. Han, Oscillation theorems for nonlinear hyperbolic systems with impulses, Nonlinear Analysis: Real World Applications, 9 (2008), 94-102.

[3]

B. T. Cui and Y. H. Yu, Oscillations of certain hyperbolic partial differential equations of neutral type, Acta Applicandae Mathematicae, 19 (1996), 80-88.

[4]

X. L. Fu and L. Q. Zhang, Force oscillation for impulsive hyperbolic boundary value problems with delay, Applied Mathematics and Computation, 158 (2004), 761-780.

[5]

M. X. He and A. P. Liu, Oscillation of hyperbolic functional differential equations, Applied Mathematics and Computation, 142 (2003), 205-224.

[6]

V. Lakshmikantham, D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.

[7]

W. N. Li, On the force oscillation of solutions for systems of impulsive parabolic differential equations with several delays, Journal of Computational and Applied Mathematics, 181 (2005), 46-57.

[8]

A. P. Liu, Oscillations of certain hyperbolic delay differential equations with damping term, Mathematics Applications, 9 (1996), 321-324.

[9]

A. P. Liu and S. C. Cao, Oscillations of the solutions of hyperbolic partial differential equations of neutral type, Chinese Quarterly Journal of Mathematics, 17 (2002), 7-13.

[10]

A. P. Liu and M. X. He, Oscillation of the solutions of nonlinear delay hyperbolic patrial differential equations of neutral type, Applied Mathematics and Mechanics, 23 (2002), 678-685.

[11]

A. P. Liu, T. Liu and M. Zou, Oscillation of nonlinear impulsive parabolic differential equations of neutral type, Rocky Mountain Journal of Mathematics, 41 (2011), 833-850.

[12]

A. P. Liu, Q. X. Ma and M. X. He, Oscillation of nonlinear impulsive parabolic equations of neutral type, Rocky Mountain Journal of Mathematics, 36 (2006), 1011-1026.

[13]

A. P. Liu, L. Xiao and T. Liu, Oscillation of nonlinear impulsive hyperbolic differential equations with several delays, Electronic Journal of Differential Equations, 24 (2004), 1-6.

[14]

J. R. Yan and C. H. Kou, Oscillation of solutions of delay impulsive differential equations, Journal of Mathematical Analysis and Applications, 254 (2001), 358-370.

[15]

M. Zou, T. Chang, Y. L. Li and A. P. Liu, Necessary and sufficient condition for oscillation of impulsive delay hyperbolic differential system, Annals of Differential Equations, 23 (2007), 608-611.

[16]

M. Zou, A. P. Liu, L. H. He and Y. L. Li, Oscillation for boundary value problem of impulsive delay parabolic differential system, Journal of Natural Science of Heilongjiang University, 23 (2006), 528-531.

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