Abstract
In this paper, we consider a class of nonlinear impulsive neutral partial functional differential equations with continuous distributed deviating arguments. For this class, we establish sufficient conditions for the H-oscillation of the solutions, using impulsive differential inequalities and an averaging technique with two different boundary conditions. We provide an example to illustrate the main result.
References
[1] M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemp. Math. Appl. 2, Hindawi Publishing, New York, 2006. 10.1155/9789775945501Search in Google Scholar
[2] J. I. Domšlak, On the oscillation of solutions of vector differential equations, Soviet Math. Dokl. 11 (1970), 839–841. Search in Google Scholar
[3] J. I. Domšlak, The oscillatoriness and non-oscillatoriness of the solutions of vector differential equations, Differ. Uravn. 7 (1971), 961–969, 1137. Search in Google Scholar
[4] L. H. Erbe, H. I. Freedman, X. Liu and J. H. Wu, Comparison principles for impulsive parabolic equations with applications to models of single species growth, J. Aust. Math. Soc. Ser. B 32 (1991), no. 4, 382–400. 10.1017/S033427000000850XSearch in Google Scholar
[5] Z. Gao and Z. Teng, Oscillation criteria of impulsive neutral parabolic equations with nonlinear diffusion coefficient, Int. J. Nonlinear Sci. 11 (2011), no. 2, 168–172. Search in Google Scholar
[6] K. Gopalsamy and B. G. Zhang, On delay differential equations with impulses, J. Math. Anal. Appl. 139 (1989), no. 1, 110–122. 10.1016/0022-247X(89)90232-1Search in Google Scholar
[7] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1988. Search in Google Scholar
[8] K. Kreith, A nonselfadjoint dynamical system, Proc. Edinburgh Math. Soc. (2) 19 (1974/75), 77–87. 10.1017/S0013091500015406Search in Google Scholar
[9] G. S. Ladde, V. Lakshmikantham and B. G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments, Monogr. Textb. Pure Appl. Math. 110, Marcel Dekker, New York, 1987. Search in Google Scholar
[10] V. Lakshmikantham, D. D. Baĭnov and P. S. Simeonov, Theory of Impulsive Differential Equations, Ser. Modern Appl. Math. 6, World Scientific, Teaneck, 1989. 10.1142/0906Search in Google Scholar
[11] W. N. Li and M. Han, Oscillation of solutions for certain impulsive vector parabolic differential equations with delays, J. Math. Anal. Appl. 326 (2007), no. 1, 363–371. 10.1016/j.jmaa.2006.03.005Search in Google Scholar
[12] W. N. Li, M. Han and F. W. Meng, H-oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments, Appl. Math. Comput. 158 (2004), no. 3, 637–653. 10.1016/j.amc.2003.10.006Search in Google Scholar
[13] G. J. Liu and C. Y. Wang, Forced oscillation of neutral impulsive parabolic partial differential equations with continuous distributed deviating arguments, Open Access Libr. J. 1 (2014), no. 9, 1–8. 10.4236/oalib.1101168Search in Google Scholar
[14] J. Luo, Oscillation of hyperbolic partial differential equations with impulses, Appl. Math. Comput. 133 (2002), no. 2–3, 309–318. 10.1016/S0096-3003(01)00217-XSearch in Google Scholar
[15] Q. Ma and A. Liu, Oscillation criteria of neutral type impulsive hyperbolic equations, Acta Math. Sci. Ser. B Engl. Ed. 34 (2014), no. 6, 1845–1853. 10.1016/S0252-9602(14)60128-4Search in Google Scholar
[16] Q. X. Ma, L. L. Zhang and A. P. Liu, Oscillation of nonlinear impulsive hyperbolic equations of neutral type, Appl. Mech. Mater. 275–277 (2013), 848–851. 10.4028/www.scientific.net/AMM.275-277.848Search in Google Scholar
[17] V. D. Mil’man and A. D. Myškis, On the stability of motion in the presence of impulses, Sibirsk. Mat. Zh. 1 (1960), 233–237. Search in Google Scholar
[18] E. Minchev and N. Yoshida, Oscillations of solutions of vector differential equations of parabolic type with functional arguments, J. Comput. Appl. Math. 151 (2003), no. 1, 107–117. 10.1016/S0377-0427(02)00740-9Search in Google Scholar
[19] E. Minchev and N. Yoshida, Oscillations of vector differential equations of hyperbolic type with functional arguments, Math. J. Toyama Univ. 26 (2003), 75–84. Search in Google Scholar
[20] E. S. Noussair and C. A. Swanson, Oscillation theorems for vector differential equations, Utilitas Math. 1 (1972), 97–109. Search in Google Scholar
[21] E. S. Noussair and C. A. Swanson, Oscillation of nonlinear vector differential equations, Ann. Mat. Pura Appl. 109 (1976), no. 1, 305–315. 10.1007/BF02416966Search in Google Scholar
[22] P. Prakash and S. Harikrishnan, Oscillation of solutions of impulsive vector hyperbolic differential equations with delays, Appl. Anal. 91 (2012), no. 3, 459–473. 10.1080/00036811.2010.541602Search in Google Scholar
[23] V. Sadhasivam and J. Kavitha, On the oscillation of solutions of fractional vector partial differential equations with deviating arguments, Amer. Rev. Math. Stat., 5 (2017), no. 1, 45–57. 10.15640/arms.v5n1a5Search in Google Scholar
[24] V. Sadhasivam, J. Kavitha and T. Raja, Forced oscillation of nonlinear impulsive hyperbolic partial differential equation with several delays, J. Appl. Math. Phys. 3 (2015), 1491–1505. 10.4236/jamp.2015.311175Search in Google Scholar
[25] V. Sadhasivam, J. Kavitha and T. Raja, Forced oscillation of impulsive neutral hyperbolic differential equations, Internat. J. Appl. Eng. Res. 11 (2016), no. 1, 58–63. Search in Google Scholar
[26] V. Sadhasivam, T. Raja and T. Kalaimani, Oscillation of nonlinear impulsive neutral functional hyperbolic equations with damping, Int. J. Pure Appl. Math. 106 (2016), no. 8, 187–197. Search in Google Scholar
[27] A. M. Samoĭlenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, River Edge, 1995. 10.1142/2892Search in Google Scholar
[28] V. S. Vladimirov, Equations of Mathematics Physics (in Russian), “Nauka”, Moscow, 1971, 10.1063/1.3022385Search in Google Scholar
[29] J. Wu, Theory and Applications of Partial Functional-Differential Equations, Appl. Math. Sci. 119, Springer, New York, 1996. 10.1007/978-1-4612-4050-1Search in Google Scholar
[30] J. Yang, A. Liu and G. Liu, Oscillation of solutions to neutral nonlinear impulsive hyperbolic equations with several delays, Electron. J. Differential Equations 2013 (2013), no. 27, 1–10. 10.1155/2013/543947Search in Google Scholar
[31] N. Yoshida, Oscillation Theory of Partial Differential Equations, World Scientific, Hackensack, 2008. 10.1142/7046Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Evaluation of some q-integrals in terms of the Dedekind eta function
- Note on the fast decay property of steady Navier–Stokes flows in the whole space
- Some uniqueness results related to the Brück conjecture
- On the oscillation of impulsive vector partial differential equations with distributed deviating arguments
Articles in the same Issue
- Frontmatter
- Evaluation of some q-integrals in terms of the Dedekind eta function
- Note on the fast decay property of steady Navier–Stokes flows in the whole space
- Some uniqueness results related to the Brück conjecture
- On the oscillation of impulsive vector partial differential equations with distributed deviating arguments