Existence results for systems of second-order impulsive differential equations
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Abstract
In this paper the authors study the existence of solutions to systems of nonlinear second order impulsive differential equations. Their results are established by using vector versions of Perov’s fixed point theorem and the nonlinear alternative of Leray-Schauder type. Both approaches are combined with a technique based on vector-valued metrics and matrices that converge to zero. Examples illustrating the results are included.
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How to Cite
Graef, J., Kadari, H., Ouahab, A., & Oumansour, A.
(2019).
Existence results for systems of second-order impulsive differential equations.
Acta Mathematica Universitatis Comenianae, 88(1), 51-66.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/679/638
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References
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[19] J. Wang and Y. Zhang, Solvability of fractional order differential systems with general nonlocal conditions, J. Appl. Math. Comput. 45 (2014), 199–222.
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[23] G. Zeng, F. Wang, and J. J. Nieto, Complexity of a delayed predator-prey model with impulsive harvest and Holling-type II functional response, Adv. Complex Syst. 11 (2008), 77–97.
[24] H. Zhang, L. S. Chen, and J. J. Nieto, A delayed epidemic model with stage structure and pulses for management strategy, Nonlinear Anal. Real World Appl. 9 (2008), 1714–1726.
[25] H. Zhang, W. Xu, and L. Chen, A impulsive infective transmission SI model for pest control, Math. Methods Appl. Sci. 30 (2007), 1169–1184.
[26] Y. Zhang and R. Wang, Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems, J. Appl. Math. Comput. 52 (2016), 323–343.
[2] O. Bolojan and R. Precup, Implicit first order differential systems with nonlocal conditions, Electron. J. Qual. Theory Differ. Equ., 2014, No. 69, 1–13.
[3] B. Dai, H. Su, and D. Hu, Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse, Nonlinear Anal. 70 (2009), 126–134.
[4] P. Georgescu, G. Morosanu, Pest regulation by means of impulsive controls, Appl. Math. Comput. 190 (2007), 790–803.
[5] J. R. Graef, J. Henderson, and A. Ouahab, Impulsive Differential Inclusions: A Fixed Point Approach, De Gruyter Series in Nonlinear Analysis and Applications 20, de Gruyter, Berlin, 2013.
[6] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[7] O. Nica, Existence results for second order three point boundary value problems, Differ. Equ. Appl. 4 (2012), 547–570.
[8] O. Nica, Initial-value problems for first-order differential systems with general nonlocal conditions, Electron. J. Differential Equations 2012 (2012), No. 74, pp. 1–15.
[9] O. Bolojan-Nica, G. Infante, and R. Precup, Existence results for systems with coupled nonlocal initial conditions, Nonlinear Anal. 4 (2014), 231–242.
[10] O. Nica and R. Precup, On the nonlocal initial value problem for first order differential systems, Stud. Univ. Babes-Bolyai Math. 56 (2011), No. 3, 125–137.
[11] J. J. Nieto and R. Rodriguez-Lopez, New comparison results for impulsive integro-differential equations and applications, J. Math. Anal. Appl. 328 (2007), 1343-1368.
[12] J. J. Nieto and R. Rodriguez-Lopez, Boundary value problems for a class of impulsive functional equations, Comput. Math. Appl. 55 (2008), 2715–2731.
[13] D. O’Regan and R. Precup, Theorems of Leray-Schauder Type and Applications, Gordon and Breach, Amsterdam, 2001.
[14] A.I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pribliz. Met. Reshen. Differ. Uravn. 2 (1964), 115–134 (in Russian).
[15] R. Precup, The role of matrices that are convergent to zero in the study of semilinear operator systems, Math. Comp. Modelling 49 (2009), 703–708.
[16] R. Precup, Methods in Nonlinear Integral Equations . Kluwer, Dordrecht, 2002.
[17] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore, 1995.
[18] J. H. Shen and J. L. Li, Existence and global attractivity of positive periodic solutions for impulsive predator-prey model with dispersion and time delays, Nonlinear Anal. Real World Appl. 10 (2009), 227–243.
[19] J. Wang and Y. Zhang, Solvability of fractional order differential systems with general nonlocal conditions, J. Appl. Math. Comput. 45 (2014), 199–222.
[20] J. R. Wang, X. Li, and Y. Zhou, Nonlocal Cauchy problems for fractional order nonlinear differential systems, Cent. Eur. J. Phys. 11, (2013), 1399–1413.
[21] W. B. Wang, J. H. Shen, and J. J. Nieto, Permanence periodic solution of predator prey system with Holling type functional response and impulses, Discrete Dyn. Nat. Soc. 2007, Art. ID 81756, 15 pp.
[22] S. T. Zavalishchin and A. N. Sesekin, Dynamic Impulse Systems, Theory and Applications, Mathematics and its Applications Vol. 394, Kluwer, Dordrecht, 1997.
[23] G. Zeng, F. Wang, and J. J. Nieto, Complexity of a delayed predator-prey model with impulsive harvest and Holling-type II functional response, Adv. Complex Syst. 11 (2008), 77–97.
[24] H. Zhang, L. S. Chen, and J. J. Nieto, A delayed epidemic model with stage structure and pulses for management strategy, Nonlinear Anal. Real World Appl. 9 (2008), 1714–1726.
[25] H. Zhang, W. Xu, and L. Chen, A impulsive infective transmission SI model for pest control, Math. Methods Appl. Sci. 30 (2007), 1169–1184.
[26] Y. Zhang and R. Wang, Nonlocal Cauchy problems for a class of implicit impulsive fractional relaxation differential systems, J. Appl. Math. Comput. 52 (2016), 323–343.