Existence and stabilization for impulsive differential equations of second order with multiple delays
DOI:
https://doi.org/10.58997/ejde.2024.07Abstract
Existence and stability of solutions are important parts in the qualitative study of delay differential equations. The stabilizing by imposing proper impulse controls are used in many areas of natural sciences and engineering. This article provides sufficient conditions for the existence and exponential stabilization of solutions to delay impulsive differential equations of second-order with multiple delays. The main tools in this article are the Schaefer fixed point theorem, fixed impulse effects, and Lyapunov-Krasovskii functionals. The outcomes extend earlier results in the literature.
For more information see https://ejde.math.txstate.edu/Volumes/2024/07/abstr.html
References
Arutyunov, Aram; Karamzin, Dmitry; Lobo Pereira, Fernando; Optimal impulsive control. The extension approach. Lecture Notes in Control and Information Sciences, 477. Springer, Cham, 2019.
Bainov, D. D.; Dimitrova, M. B.; Sufficient conditions for the oscillation of bounded solutions of a class of impulsive differential equations of second order with a constant delay. Georgian Math. J., 6 (1999), no. 2, 99-106.
Bainov, D. D.; Simeonov, P. S.; Impulsive differential equations. Asymptotic properties of the solutions. Translated from the Bulgarian manuscript by V. Covachev[V. Khr. Kovachev]. Series on Advances in Mathematics for Applied Sciences, 28. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
Belfo, Joao P.; Lemos, Joao M.; Optimal impulsive control for cancer therapy. Springer Briefs in Electrical and Computer Engineering. SpringerBriefs in Control, Automation and Robotics. Springer, Cham, 2021.
Benchohra, M.; Henderson, J.; Ntouyas, S., Ouahabi; A., Higher order impulsive functional differential equations with variable times. Dynam. Systems Appl. 12 (2003), no. 3-4, 383-392.
Benchohra, M.; Henderson, J.; Ntouyas, S.; Impulsive differential equations and inclusions. Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006.
Columbu, A.; Frassu, S.; Viglialoro, G.; Properties of given and detected unbounded solutions to a class of chemotaxis models. Stud. Appl. Math., 151 (2023), no. 4, 1349-1379.
Feng, Wei Zhen; Impulsive stabilization for second-order differential equations. J. South China Normal Univ. Natur. Sci. Ed., 2001, no. 1, 16-19.
Gimenes, L. P.; Federson, M.; Existence and impulsive stability for second order retarded differential equations. Appl. Math. Comput., 177 (2006), no. 1, 44-62.
Gimenes, L. P.; Federson, M.; Taboas, P.; Impulsive stability for systems of second order retarded differential equations. Nonlinear Anal., 67 (2007), no. 2, 545-553.
Graef, J. R.; Kadari, H.; Ouahab, A.; Oumansour, A.; Existence results for systems of secondorder impulsive differential equations. Acta Math. Univ. Comenian. (N.S.), 88 (2019), no. 1, 51-66.
Graef, J. R., Tunžc, C.; Continuability and boundedness of multi-delay functional integro-differential equations of the second order. RACSAM 109, 169-173 (2015). https://doi.org/10.1007/s13398-014-0175-5
Huang, C.; Liu, B.; Qian, C.; Cao, J.; Stability on positive pseudo almost periodic solutions of HPDCNNs incorporating D operator. Math. Comput. Simulation, 190 (2021), 1150-1163.
Huang, C.; Liu, B.; Yang, H.; Cao, J.; Positive almost periodicity on SICNNs incorporating mixed delays and D operator. Nonlinear Anal. Model. Control, 27 (2022), no. 4, 719-739.
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.
Li, Hua; Luo, Zhiguo; Boundedness results for impulsive functional differential equations with infinite delays. J. Appl. Math. Comput., 18 (2005), no. 1-2, 261-272.
Li, T.; Frassu, S.; Viglialoro, G.; Combining effects ensuring boundedness in an attractionrepulsion chemotaxis model with production and consumption. Z. Angew. Math. Phys., 74 (2023), no. 3, Paper No. 109, 21 pp.
Li, T.; Pintus, N.; Viglialoro, G.; Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys., 70 (2019), no. 3, Paper No. 86, 18 pp.
Li, T.; Viglialoro, G.; Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differential Integral Equations, 34 (2021), no. 5-6, 315-336.
Li, X.; Bohner, M.; Wang, C.-K.; Impulsive differential equations: periodic solutions and applications. Automatica J. IFAC, 52 (2015), 173-178.
Li, Xian; Weng, Peixuan; Impulsive stabilization of two kinds of second-order linear delay differential equations. J. Math. Anal. Appl., 291 (2004), no. 1, 270-281.
Li, Xiaodi; Uniform asymptotic stability and global stability of impulsive infinite delay differential equations. Nonlinear Anal., 70 (2009), no. 5, 1975-1983.
Li, Xiaodi; New results on global exponential stabilization of impulsive functional differential equations with infinite delays or finite delays. Nonlinear Anal. Real World Appl. 11 (2010), no. 5, 4194-4201.
Li, Xiaodi; Further analysis on uniform stability of impulsive infinite delay differential equations. Appl. Math. Lett. 25 (2012), no. 2, 133-137.
Li, Xiaodi; Song, Shiji; Impulsive systems with delays-stability and control. Springer, Singapore; Science Press Beijing, Beijing, 2022.
Li, Zhengguo; Soh, Yengchai; Wen, Changyun; Switched and impulsive systems. Analysis, design, and applications. Lecture Notes in Control and Information Sciences, 313. Springer- Verlag, Berlin, 2005.
Liu, Juan; Li, Xiaodi; Impulsive stabilization of high-order nonlinear retarded differential equations. Appl. Math. 58 (2013), no. 3, 347-367.
Luo, Zhiguo; Shen, J.; Impulsive stabilization of functional differential equations with infinite delays. Appl. Math. Lett., 16 (2003), no. 5, 695-701.
Pandit, S. G.; Deo, Sadashiv G.; Differential systems involving impulses. Lecture Notes in Mathematics, 954. Springer-Verlag, Berlin-New York, 1982.
Pinelas, S.; Tunžc, O.; Solution estimates and stability tests for nonlinear delay integrodifferential equations, Electron. J. Differential Equations(2022), Paper No. 68, 12 pp.
Samoilenko, A. M.; Perestyuk, N. A.; Impulsive differential equations. With a preface by Yu. A. Mitropolskii and a supplement by S. I. Trofimchuk. Translated from the Russian by Y. Chapovsky.World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 14. World Scientific Publishing Co., Inc., River Edge, NJ, 1995.
Smart, D. R.; Fixed point theorems, Cambridge Tracts in Mathematics, No. 66. Cambridge University Press, London-New York, 1974.
Stamova, I.; Stability analysis of impulsive functional differential equations. De Gruyter Expositions in Mathematics, 52. Walter de Gruyter GmbH & Co. KG, Berlin, 2009.
Stamova, I.,. Stamov, G.; Applied impulsive mathematical models. CMS Books in Mathematics/ Ouvrages de Mathematiques de la SMC. Springer, Cham, 2016.
Stamova, I. M., Stamov, G. T.; Functional and impulsive differential equations of fractional order. Qualitative analysis and applications. CRC Press, Boca Raton, FL, 2017.
Tunžc, C.; Tunžc, O.; On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 115 (2021), no. 3, Paper No. 115, 17 pp. https://doi.org/10.1007/s13398-021-01058-8
Tunžc, C.; Tunžc, O.; On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order. Mathematics, 2022; 10(22):4235. https://doi.org/10.3390/math10224235
Tunžc, C.; Tunžc, O.; Ulam stabilities of nonlinear iterative integro-differential equations. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 117, 118 (2023). https://doi.org/10.1007/s13398-023-01450-6
Tunžc, O.; On the fundamental analyses of solutions to nonlinear integro-differential equations of second order. J. Nonlinear Convex Anal., 24 (2023), no. 1, 17-32.
Tunžc, O.; Tunžc, C.; Wen C.-F., Yao, J.-C.; On the qualitative analyses solutions of new mathematical models of integro-differential equations with infinite delay. Math. Meth. Appl. Sci. (2023), 1-17. https://doi.org/10.1002/mma.9306
Tunžc, O.; Tunžc, C.; Solution estimates to Caputo proportional fractional derivative delay integro-differential equations. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM, 117 (2023), no. 1, Paper No. 12, 13 pp. https://doi.org/10.1007/s13398-022-01345-y
Tunžc, C.; Tunžc, O.; Yao, J.-C.; On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations. Symmetry 2023, 15, 109. https://doi.org/10.3390/sym15010109
Tunžc, O.; Tunžc, C; Yao, J.-C.; On the existence of results for multiple retarded differential and integro-differential equations of second order. J. Nonlinear Convex Anal., 25 (2024), (accepted).
Tunžc, C., Wang, Y.; Tunžc, O.; Yao, J.-C.; New and Improved Criteria on Fundamental Properties of Solutions of Integro-Delay Differential Equations with Constant Delay. Mathematics. 2021; 9(24):3317. https://doi.org/10.3390/math9243317
Xie, S. L.; Existence of solutions to damped second-order impulsive functional differential equations with infinite delay. (Chinese) Acta Math. Sci. Ser. A (Chinese Ed.) 35 (2015), no. 1, 97-109.
Wen, Q.; Ren, L.; Liu, R.; Existence and uniqueness of periodic solution to second-order impulsive differential equations. Math. Methods Appl. Sci. 46 (2023), no. 5, 6191-6209.
Weng, A., Sun, J.; Impulsive stabilization of second-order delay differential equations. Nonlinear Anal. Real World Appl. 8 (2007), no. 5, 1410-1420.
Weng, A., Sun, J.; Impulsive stabilization of second-order nonlinear delay differential systems. Appl. Math. Comput. 214 (2009), no. 1, 95-101.
Yang, T.; Impulsive control theory. Lecture Notes in Control and Information Sciences, 272. Springer-Verlag, Berlin, 2001.
Zhang, Y., Sun, J.; Boundedness of the solutions of impulsive differential systems with timevarying delay. Appl. Math. Comput. 154 (2004), no. 1, 279-288.
Zhao, X.; Liu, B.; Qian, C.; Cao, J.; Stability analysis of delay patchconstructed Nicholsons blowflies system, Math. Comput. Simulation (2023), (in press). https://doi.org/10.1016/j.matcom.2023.09.012
Downloads
Published
License
Copyright (c) 2024 Sandra Pinelas, Osman Tunç, Erdal Korkmaz, Cemil Tunç
This work is licensed under a Creative Commons Attribution 4.0 International License.
