Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays

Affiliation(s)

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, India.

Post Graduate and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, India.

ABSTRACT

In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.

In this paper, we study oscillatory properties of solutions for the nonlinear impulsive hyperbolic equations with several delays. We establish sufficient conditions for oscillation of all solutions.

Cite this paper

Sadhasivam, V. , Kavitha, J. and Raja, T. (2015) Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays.*Journal of Applied Mathematics and Physics*, **3**, 1491-1505. doi: 10.4236/jamp.2015.311175.

Sadhasivam, V. , Kavitha, J. and Raja, T. (2015) Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays.

References

[1] Wu, J.H. (1996) Theory of Partial Functional Differential Equations and Applications. New York, Springer.

http://dx.doi.org/10.1007/978-1-4612-4050-1

[2] Liu, A.P. (1996) Oscillations of Certain Hyperbolic Delay Differential Equations with Damping Term. Mathematica Applicate, 9, 321-324.

[3] Liu, A.P., Xiao, L. and Liu, T. (2002) Oscillations of the Solutions of Hyperbolic Partial Functional Differential Equations of Neutral Type. Acta Analysis Functionalis Applicate, 4, 69-74.

[4] He, M.X. and Liu, A.P. (2003) Oscillation of Hyperbolic Partial Differential Equations. Applied Mathematics and Computation, 142, 205-224.

http://dx.doi.org/10.1016/S0096-3003(02)00295-3

[5] Shoukaku, Y. (2011) Oscillation of Solutions for Forced Nonlinear Neutral Hyperbolic Equations with Functional Arguments. Electronic Journal of Differential Equations, 2011, 1-16.

[6] Shoukaku, Y. and Yoshida, N. (2010) Oscillation of Nonlinear Hyperbolic Equations with Functional Arguments via Riccati Method. Applied Mathematics and Computation, 217, 143-151.

http://dx.doi.org/10.1016/j.amc.2010.05.030

[7] Yoshida, N. (2008) Oscillation Theory of Partial Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/7046

[8] Erbe, L., Freedman, H., Liu, X.Z. and Wu, J.H. (1991) Comparison Principles for Impulsive Parabolic Equations with Application to Models of Single Species Growth. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 382-400.

http://dx.doi.org/10.1017/S033427000000850X

[9] Bainov, D.D. and Simeonov, P.S. (1989) Systems with Impulse Effect: Stability Theory and Applications. Ellis Horwood, Chichester.

[10] Bainov, D.D. and Simeonov, P.S. (1993) Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow.

[11] Chan, C. and Ke, L. (1994) Remarks on Impulsive Quenching Problems. Proceedings of Dynamics Systems and Applications, 1, 59-62.

[12] Zhang, L.Q. (2000) Oscillation Criteria for Hyperbolic Partial Differential Equations with Fixed Moments of Impulse Effects. Acta Mathematica Sinica, 43, 17-26.

[13] Mil’man, V.D. and Myshkis, A.D. (1960) On the Stability of Motion in the Presence of Impulses. Siberian Mathematical Journal, 1, 233-237.

[14] Bainov, D.D., Kamont, Z. and Minchev, E. (1996) Monotone Iterative Methods for Impulsive Hyperbolic Differential Functional Equations. Journal of Computational and Applied Mathematics, 70, 329-347.

http://dx.doi.org/10.1016/0377-0427(95)00209-X

[15] Cui, B.T., Han, M.A. and Yang, H.Z. (2005) Some Sufficient Conditions for Oscillation of Impulsive Delay Hyperbolic Systems with Robin Boundary Conditions. Journal of Computational and Applied Mathematics, 180, 365-375.

http://dx.doi.org/10.1016/j.cam.2004.11.006

[16] Cui, B.T., Liu, Y.Q. and Deng, F.Q. (2003) Some Oscillation Problems for Impulsive Hyperbolic Differential Systems with Several Delays. Applied Mathematics and Computation, 146, 667-679.

http://dx.doi.org/10.1016/S0096-3003(02)00611-2

[17] Deng, L.H. and Ge, W.G. (2001) Oscillation Criteria of Solutions for Impulsive Delay Parabolic Equations. Acta Mathematica Sinica, 44, 501-506.

[18] Yang, J.C., Liu, A.P. and Liu, G.J. (2013) Oscillation of Solutions to Neutral Nonlinear Impulsive Hyperbolic Equations with Several Delays. Electronic Journal of Differential Equations, 2013, 1-10.

[19] Fu, X.L. and Shiau, L.J. (2013) Oscillation Criteria for Impulsive Parabolic Boundary Value Problem with Delay. Applied Mathematics and Computation, 153, 587-599.

[20] Ye, Q.X. and Li, Z.Y. (1990) Introduction to Reaction Diffusion Equations. Science Press, Beijing.

[21] Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S. (1989) Theory of Impulsive Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/0906

[22] Luo, J.W. (2002) Oscillation of Hyperbolic Partial Differential Equations with Impulses. Applied Mathematics and Computation, 133, 309-318.

http://dx.doi.org/10.1016/S0096-3003(01)00217-X

[1] Wu, J.H. (1996) Theory of Partial Functional Differential Equations and Applications. New York, Springer.

http://dx.doi.org/10.1007/978-1-4612-4050-1

[2] Liu, A.P. (1996) Oscillations of Certain Hyperbolic Delay Differential Equations with Damping Term. Mathematica Applicate, 9, 321-324.

[3] Liu, A.P., Xiao, L. and Liu, T. (2002) Oscillations of the Solutions of Hyperbolic Partial Functional Differential Equations of Neutral Type. Acta Analysis Functionalis Applicate, 4, 69-74.

[4] He, M.X. and Liu, A.P. (2003) Oscillation of Hyperbolic Partial Differential Equations. Applied Mathematics and Computation, 142, 205-224.

http://dx.doi.org/10.1016/S0096-3003(02)00295-3

[5] Shoukaku, Y. (2011) Oscillation of Solutions for Forced Nonlinear Neutral Hyperbolic Equations with Functional Arguments. Electronic Journal of Differential Equations, 2011, 1-16.

[6] Shoukaku, Y. and Yoshida, N. (2010) Oscillation of Nonlinear Hyperbolic Equations with Functional Arguments via Riccati Method. Applied Mathematics and Computation, 217, 143-151.

http://dx.doi.org/10.1016/j.amc.2010.05.030

[7] Yoshida, N. (2008) Oscillation Theory of Partial Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/7046

[8] Erbe, L., Freedman, H., Liu, X.Z. and Wu, J.H. (1991) Comparison Principles for Impulsive Parabolic Equations with Application to Models of Single Species Growth. The Journal of the Australian Mathematical Society. Series B. Applied Mathematics, 32, 382-400.

http://dx.doi.org/10.1017/S033427000000850X

[9] Bainov, D.D. and Simeonov, P.S. (1989) Systems with Impulse Effect: Stability Theory and Applications. Ellis Horwood, Chichester.

[10] Bainov, D.D. and Simeonov, P.S. (1993) Impulsive Differential Equations: Periodic Solutions and Applications. Longman, Harlow.

[11] Chan, C. and Ke, L. (1994) Remarks on Impulsive Quenching Problems. Proceedings of Dynamics Systems and Applications, 1, 59-62.

[12] Zhang, L.Q. (2000) Oscillation Criteria for Hyperbolic Partial Differential Equations with Fixed Moments of Impulse Effects. Acta Mathematica Sinica, 43, 17-26.

[13] Mil’man, V.D. and Myshkis, A.D. (1960) On the Stability of Motion in the Presence of Impulses. Siberian Mathematical Journal, 1, 233-237.

[14] Bainov, D.D., Kamont, Z. and Minchev, E. (1996) Monotone Iterative Methods for Impulsive Hyperbolic Differential Functional Equations. Journal of Computational and Applied Mathematics, 70, 329-347.

http://dx.doi.org/10.1016/0377-0427(95)00209-X

[15] Cui, B.T., Han, M.A. and Yang, H.Z. (2005) Some Sufficient Conditions for Oscillation of Impulsive Delay Hyperbolic Systems with Robin Boundary Conditions. Journal of Computational and Applied Mathematics, 180, 365-375.

http://dx.doi.org/10.1016/j.cam.2004.11.006

[16] Cui, B.T., Liu, Y.Q. and Deng, F.Q. (2003) Some Oscillation Problems for Impulsive Hyperbolic Differential Systems with Several Delays. Applied Mathematics and Computation, 146, 667-679.

http://dx.doi.org/10.1016/S0096-3003(02)00611-2

[17] Deng, L.H. and Ge, W.G. (2001) Oscillation Criteria of Solutions for Impulsive Delay Parabolic Equations. Acta Mathematica Sinica, 44, 501-506.

[18] Yang, J.C., Liu, A.P. and Liu, G.J. (2013) Oscillation of Solutions to Neutral Nonlinear Impulsive Hyperbolic Equations with Several Delays. Electronic Journal of Differential Equations, 2013, 1-10.

[19] Fu, X.L. and Shiau, L.J. (2013) Oscillation Criteria for Impulsive Parabolic Boundary Value Problem with Delay. Applied Mathematics and Computation, 153, 587-599.

[20] Ye, Q.X. and Li, Z.Y. (1990) Introduction to Reaction Diffusion Equations. Science Press, Beijing.

[21] Lakshmikantham, V., Bainov, D.D. and Simeonov, P.S. (1989) Theory of Impulsive Differential Equations. World Scientific, Singapore.

http://dx.doi.org/10.1142/0906

[22] Luo, J.W. (2002) Oscillation of Hyperbolic Partial Differential Equations with Impulses. Applied Mathematics and Computation, 133, 309-318.

http://dx.doi.org/10.1016/S0096-3003(01)00217-X