On the Oscillation of Conformable Impulsive Vector Partial Differential Equations

Chatzarakis, George E.; Logaarasi, Kandhasamy; Raja, Thangaraj; Sadhasivam, Vadivel

doi:10.2478/tmmp-2020-0021

Abstract

In this paper, we study the oscillations of a class of conformable impulsive vector partial functional differential equations. For this class, our approach is to reduce the multi-dimensional oscillation problems to that of one dimensional impulsive delay differential inequalities by applying inner product reducing dimension method and an impulsive differential inequality technique. We provide an example to illustrate the effectiveness of our main results.

References

[1] ATANGANA, A.—BALEANU, D.—ALSAEDI, A.: New properties of conformable derivative, Open Math. 13 (2015), 889–898.10.1515/math-2015-0081
Search in Google Scholar Back to article
[2] BENCHOHRA, M.—HENDERSON, J.—NTOUYAS, S.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York, 2006.10.1155/9789775945501
Search in Google Scholar Back to article
[3] CHATZARAKIS, G. E.—LOGAARASI, K.—RAJA, T.—SADHASIVAM, V.: Interval oscillation criteria for conformable fractional differential equations with impulses, Appl. Math. E-Notes 19 (2019), 354–369.
Search in Google Scholar Back to article
[4] CHATZARAKIS, G. E.—SADHASIVAM, V.—RAJA, T.: On the oscillation of impulsive vector partial differential equations with distributed deviating arguments, Analysis (Berlin) 38 (2018), no. 2, 101–114.
Search in Google Scholar Back to article
[5] CHATZARAKIS, G.-E.—SADHASIVAM, V.—RAJA, T. —KALAIMANI, T.: Oscillation of certain nonlinear impulsive neutral partial differential equations with continuous distributed deviating arguments and a damping term, Dynam. Contin. Discrete Impuls. Systems 25 (2018), 329–345.
Search in Google Scholar Back to article
[6] CHEN, D.-X.: Oscillation criteria of fractional differential equations, Adv. Difference Equ. 2012 (2012), no. 33, 10 pp.
Search in Google Scholar Back to article
[7] GRACE, S.-R.—AGARWAL, R.-P.—WONG, P.-J.-Y.—ZAFER, A.: On the oscillation of fractional differential equations, Frac. Calc. Appl. Anal. 15 (2012), 222–231.10.2478/s13540-012-0016-1
Search in Google Scholar Back to article
[8] DOMŠLAK, JU.-I.: On the oscillation of solutions of vector differential equations, Soviet Math. Dokl. 11 (1970), 839–841.
Search in Google Scholar Back to article
[9] CHIU, K.-S.—LI, T.: Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments, Math. Nachr. 292 (2019), no. 10, 2153–2164.
Search in Google Scholar Back to article
[10] ERBE, L.-H.—FREEDMAN, H.-I.—LIU, X.-Z.—WU, J.-H.: Comparison principles for impulsive parabolic equations with application to models of single species growth, J. Austral. Math. Soc. Ser (B). 32 (1991), no. 4, 382–400.
Search in Google Scholar Back to article
[11] GAO, Z.—TENG, Z.: 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 Back to article
[12] GOPALSAMY, K.—ZHANG, B.-G.: On delay differential equations with impulses, J. Math. Anal. Appl. 139 (1989), 110–122.10.1016/0022-247X(89)90232-1
Search in Google Scholar Back to article
[13] HARDY, G. H.—LITTLEWOOD, J. E.—PÓLYA, G.: Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, UK, 1988.
Search in Google Scholar Back to article
[14] KALAIMANI, T.—RAJA, T.—SADHASIVAM, V.—SAKER, S. H.: Oscillation of impulsive neutral partial differential equations with distributed deviating arguments, Bul. Math. Soc. Sci. Math. Roumanie (N.S.) 61 (2018), no. 109, 51–68.
Search in Google Scholar Back to article
[15] KHALIL, R. R.—AL HORANI, M.—YOUSEF, A.—SABABHEH, M.: A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70.10.1016/j.cam.2014.01.002
Search in Google Scholar Back to article
[16] KILBAS, A. A.—SRIVASTAVA, H. M.—TRUJILLO, J. J.: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam, 2006.
Search in Google Scholar Back to article
[17] LADDE, G. S.—LAKSHMIKANTHAM, V.—ZHANG, B. G.: Oscillation Theory of Differential Equations with Deviating Arguments. In: Monographs and Textbooks in Pure and Applied Mathematics Vol. 110. Marcel Dekker, Inc., New York, 1987.
Search in Google Scholar Back to article
[18] LAKSHMIKANTHAM, V.—BAINOV, D. D.—SIMEONOV, P. S.: Theory of Impulsive Differential Equations. In: Series in Modern Applied Mathematics Vol. 6. World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.
Search in Google Scholar Back to article
[19] LI, W. N.—HAN, M.: Oscillation of solutions for certain impulsive vector parabolic differential equations with delays, J. Math. Anal. Appl., 326(1) (2007), 363–371.10.1016/j.jmaa.2006.03.005
Search in Google Scholar Back to article
[20] LI, W. N.—HAN, M.—MENG, F. W.: H-oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments, Appl. Math. Comput., 158 (2004), 637–653.
Search in Google Scholar Back to article
[21] LIU, G. J.—WANG, C. Y. : Forced oscillation of neutral impulsive parabolic partial differential equations with continuous distributed deviating arguments, Open Access Library Journal 1 (2014), no. 9, 1–8.
Search in Google Scholar Back to article
[22] MA, Q. X.—LIU, A. P.: Oscillation criteria of neutral type impulsive hyperbolic equations, Acta Math. Sci. 34 B (2014), no. 6, 1845–1853.
Search in Google Scholar Back to article
[23] MIL’MAN, V. D.—MYŠKIS, A. D.: On the stability of motion in the presence of impulse, Sibirsk. Math. Ž., 1 (1960), no. 2, 233–237.
Search in Google Scholar Back to article
[24] MINCHEV, E.—YOSHIDA, N.: Oscillation of vector differential equations of hyperbolic type with functional arguments, Math. J. Toyama Univ. 26 (2003), 75–84.
Search in Google Scholar Back to article
[25] MINCHEV, E.—YOSHIDA, N.: Oscillation of solutions of vector differential equations of parabolic type with functional arguments, J. Comput. Appl. Math., 151(1) (2003), no. 1, 107–117.
Search in Google Scholar Back to article
[26] PODLUBNY, I.: Fractional Differential Equations. Academic Press, San Diego, 1999.
Search in Google Scholar Back to article
[27] PRAKASH, P.—HARIKRISHNAN, S.: Oscillation of solutions of impulsive vector hyperbolic differential equations with delays, Appl. Anal. 91 (2012), no. 3, 459–473.
Search in Google Scholar Back to article
[28] SADHASIVAM, V.—LOGAARASI, K.—KALAIMANI, T.: Oscillation of even order impulsive partial differential equations with deviating arguments, Int. J. Pure Appl. Math., 115(9) (2017), 83–91.
Search in Google Scholar Back to article
[29] SAMOILENKO, A. M.—PERSESTYUK, N. A.: Impulsive Differential Equations. World Scientific, Singapore, 1995.10.1142/2892
Search in Google Scholar Back to article
[30] TANG, S.—ZADA, A.—FAISAL, S.—EL SHEIKH, M. M. A.—LI, T.: Stability of higher-order nonlinear impulsive differential equations, J. Nonlinear Sci. Appl., 9 (2016), no. 6, 4713–4721.
Search in Google Scholar Back to article
[31] WANG, P.—LI, C.—ZHANG, J.—LI, T.: Quasilinearization method for first-order impulsive integro differential equations, Electron. J. Differential Equations, 2019 (2019), 1–14.10.1155/2019/3238462
Search in Google Scholar Back to article
[32] WU, J. H.: Theory and Applications of Partial Functional Differential Equations. In: Applied Mathematical Sciences Vol. 119. Springer-Verlag, New York, 1996.
Search in Google Scholar Back to article
[33] YOSHIDA, N.: Oscillation Theory of Partial Differential Equations. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.10.1142/7046
Search in Google Scholar Back to article

On the Oscillation of Conformable Impulsive Vector Partial Differential Equations

Abstract

Paradigm

My account