Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter November 20, 2018

Sturm-Picone comparison theorems for nonlinear impulsive differential equations

  • Zeynep Kayar EMAIL logo and Sarbast Kamal Rasheed Masiha
From the journal Mathematica Slovaca

Abstract

The celebrated Sturm-Picone comparison theorem as well as the well known Leighton’s variational lemma and Leighton’s theorem, all of which are the fundamental tools of comparison and so, oscillation theory, are obtained for regular and singular nonlinear impulsive differential equations and related previous results in the literature are generalized to such equations.

  1. (Communicated by Michal Fečkan)

References

[1] Akhmet, M. U.: The complex dynamics of the cardiovascular system, Nonlinear Anal. 71 (2009), 1922–1931.10.1016/j.na.2009.02.103Search in Google Scholar

[2] Akhmet, M. U.—Bekmukhambetova, G. A.: A prototype compartmental model of blood pressure distribution, Nonlinear Anal. Real World Appl. 11 (2010), 1249–1257.10.1016/j.nonrwa.2009.02.015Search in Google Scholar

[3] Akhmet, M.: Principles of Discontinuous Dynamical Systems, Springer, New York, 2010.10.1007/978-1-4419-6581-3Search in Google Scholar

[4] Bainov, D. D. et al: Sturmian comparison theory for impulsive differential inequalities and equations, Arch. Math. (Basel) 67 (1996), 35–49.10.1007/BF01196165Search in Google Scholar

[5] Dong, L. et al: Optimal harvesting policy for inshore-offshore fishery model with impulsive diffusion, Acta Math. Sci. Ser. B Engl. Ed. 27 (2007), 405–412.10.1016/S0252-9602(07)60040-XSearch in Google Scholar

[6] Došlý, O.—řehák, P.: Half Linear Differential Equations. North-Holland Mathematics Studies 202, Elsevier Science B.V., Amsterdam, 2005.Search in Google Scholar

[7] Huang, M. et al: Modeling impulsive injections of insulin: towards artificial pancreas, SIAM J. Appl. Math. 72 (2012), 1524–1548.10.1137/110860306Search in Google Scholar

[8] Jaroš, J.—Kusano, T.: A Picone type identity for second order half-linear differential equations, Acta Math. Univ. Comenian. (N.S.) 68 (1999), 137–151.Search in Google Scholar

[9] Kreith, K.: Oscillation Theory. Lecture Notes in Math. 324, Springer-Verlag, 1973.10.1007/BFb0067537Search in Google Scholar

[10] Kreith, K.: A Picone identity for first order differential systems, J. Math. Anal. Appl. 31 (1970), 297–308.10.1016/0022-247X(70)90024-7Search in Google Scholar

[11] Lakshmikantham, V.—Bainov, D. D.—Simeonov, P. S.: Theory of Impulsive Differential Equations. Series in Modern Applied Mathematics 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989.10.1142/0906Search in Google Scholar

[12] Leighton, W.: Comparison theorems for linear differential equations of second order, Proc. Amer. Math. Soc. 13 (1962), 603–610.10.1090/S0002-9939-1962-0140759-0Search in Google Scholar

[13] Li, X.—Rakkiyappan, R.: Impulse controller design for exponential synchronization of chaotic neural networks with mixed delays, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), 1515–1523.10.1016/j.cnsns.2012.08.032Search in Google Scholar

[14] Li, H. J.—Yeh, C. C.: Sturmian comparison theorem for half-linear second-order differential equations, Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 1193–1204.10.1017/S0308210500030468Search in Google Scholar

[15] Lisena, B.: Dynamical behavior of impulsive and periodic Cohen-Grossberg neural networks, Nonlinear Anal. 74(13) (2011), 4511–4519.10.1016/j.na.2011.04.015Search in Google Scholar

[16] Miron, R. E.—Smith, R. J.: Resistance to protease inhibitors in a model of HIV-1 infection with impulsive drug effects, Bull. Math. Biol. 76 (2014), 59–97.10.1007/s11538-013-9903-9Search in Google Scholar PubMed

[17] Muller-Pfeiffer, E.: Sturm comparison theorems for nonselfadjoint second order differential equations on noncompact intervals, Math. Nachr. 159 (1992), 291–298.10.1002/mana.19921590120Search in Google Scholar

[18] Özbekler, A.—Zafer, A.: Sturmian comparison theory for linear and half-linear impulsive differential equations, Nonlinear Anal. 63 (2005), 289 – 297.10.1016/j.na.2005.01.087Search in Google Scholar

[19] Özbekler, A.—Zafer, A.: Picone’s formula for linear non-selfadjoint impulsive differential equations, J. Math. Anal. Appl. 319 (2006), 410–423.10.1016/j.jmaa.2005.06.019Search in Google Scholar

[20] Özbekler, A.—Zafer, A.: Picone type formula for non-selfadjoint impulsive differential equations with discontinuous solutions, Electron. J. Qual. Theory Differ. Equ. 2010 (2010), 12.10.14232/ejqtde.2010.1.35Search in Google Scholar

[21] Özbekler, A.: Picone type formula for half-linear impulsive differential equations with discontinuous solutions, Math. Methods Appl. Sci. 38 (2015), 1592–1600.10.1002/mma.3171Search in Google Scholar

[22] Picone, M.: Sui valori eccezionali di un parametro da cui dipende unequazione differenziale lineare ordinaria del second ordine, Ann. Scuola Norm. Pisa 11 (1909), 1–141.Search in Google Scholar

[23] Pilipchuk, V. N.—Ibrahim, R. A.: Dynamics of a two-pendulum model with impact interaction and an elastic support, Nonlinear Dynam. 21(3) (2000), 221–247.10.1023/A:1008333123695Search in Google Scholar

[24] Reid, W. T.: Sturmian Theory for Ordinary Differential Equations. Appl. Math. Sci. 31, Springer-Verlag, New York-Berlin, 1980.10.1007/978-1-4612-6110-0Search in Google Scholar

[25] Samoilenko, A. M.—Perestyuk, N. A.: Impulsive Differential Equations. World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises 14, World Scientific Publishing Co. Inc., River Edge, NJ, 1995.10.1142/2892Search in Google Scholar

[26] Stamova, I. M.—Emmenegger, J. F.: Stability of the solutions of impulsive functional differential equations modelling price fluctuations in single commodity markets, Int. J. Appl. Math. 15 (2004), 271–290.Search in Google Scholar

[27] Sturm, C.: Sur les équations difféntielles linéaries du second ordere, J. Math. Pures. Appl. 1 (1836), 106–186.Search in Google Scholar

[28] Sun, S.—Chen, L.: Mathematical modelling to control a pest population by infected pests, Appl. Math. Model. 33 (2009), 2864–2873.10.1016/j.apm.2008.08.018Search in Google Scholar

[29] Swanson, C. A.: Comparison and Oscillation Theory of Linear Differential Equations, Mathematics in Science and Engineering 48, Academic Press, New York-London, 1968.Search in Google Scholar

[30] Tiryaki, A.: Sturm-Picone type theorems for second-order nonlinear differential equations, Electron. J. Differential Equations 2014 (2014), 1–10.10.14232/ejqtde.2014.1.46Search in Google Scholar

[31] Tyagi, J.: Generalizations of Sturm-Picone theorem for second-order nonlinear differential equations, Taiwanese J. Math. 17 (2013), 361–378.10.11650/tjm.17.2013.2074Search in Google Scholar

Received: 2017-04-10
Accepted: 2017-09-18
Published Online: 2018-11-20
Published in Print: 2018-12-19

© 2018 Mathematical Institute Slovak Academy of Sciences

Downloaded on 18.4.2024 from https://www.degruyter.com/document/doi/10.1515/ms-2017-0188/html
Scroll to top button