The averaging method of set impulsive differential equations with initial and boundary value conditions. (Chinese. English summary) Zbl 1513.34168
Summary: This paper uses the scheme of full, and partially additive averaging method to study the set impulsive differential equations with the initial and multipoint boundary value problems in Euclidean space \(\mathbb{R}^n\), and proves the approximate relationship of the solutions between the original equations and the average equations.
MSC:
| 34C29 | Averaging method for ordinary differential equations |
| 34B37 | Boundary value problems with impulses for ordinary differential equations |
| 34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |
| 34A60 | Ordinary differential inclusions |
References:
| [1] | Bainov D , Domshlak Y , Milusheva S . Partial averaging for impulsive differential equations with supremum. Georgian Mathematical Journal, 1996, 3 (1): 11- 26 · Zbl 0837.34018 · doi:10.1515/GMJ.1996.11 |
| [2] | Bainov D , Milusheva S . Application of the averaging method for functional-differential equations with impulses. J Math Anal Appl, 1983, 95, 85- 105 · Zbl 0522.34071 · doi:10.1016/0022-247X(83)90137-3 |
| [3] | Bainov D , Simeonov P . Impulsive Differential Equations. Singapore: World Scientific, 1995 · Zbl 0828.34002 |
| [4] | Bainov D , Simeonov P . Systems with Impulsive Effect: Stability, Theory and Applications. New York: Halsted Press, 1998 · Zbl 0949.34002 |
| [5] | Benchohra M , Boucherif A . An existence result for first order initial value problems for impulsive differential inclusions in Banach spaces. Archivum Mathematicum (Brno), 2000, 36, 159- 169 · Zbl 1054.34099 |
| [6] | Devi J V . Basic results in impulsive set differential equations. Nonlinear Studies, 2003, 10 (3): 259- 271 · Zbl 1055.34008 |
| [7] | Devi J V , Vatsala A S . A study of set differential equations with delay. Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical Analysis, 2004, 11, 287- 300 · Zbl 1069.34112 |
| [8] | Federson M , Mesquita J G . Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses. J Differential Equations, 2013, 255, 3098- 3126 · Zbl 1325.34085 · doi:10.1016/j.jde.2013.07.026 |
| [9] | Kitanov N . Method of averaging for optimal control problems with impulsive effects. Inter J Pure Appl Math, 2011, 72 (4): 573- 589 · Zbl 1242.49075 |
| [10] | Klymchuk S , Plotnikov A , Skripnik N . Overview of V.A. Plotnikov’s research on averaging of differential inclusions. Physica D, 2012, 241 (22): 1932- 1947 · doi:10.1016/j.physd.2011.05.004 |
| [11] | Liu J , Zhang Z X , Jiang W . Global mittag-leffler stability of fractional order nonlinear impulsive differential systems with time delay. Acta Math Sci, 2020, 40A (4): 1053- 1063 · Zbl 1474.34539 · doi:10.3969/j.issn.1003-3998.2020.04.020 |
| [12] | Lakshmikantham V , Bainov D , Simeonov P . Theory of Impulsive Differential Equations. Singapore: World Scientific, 1989 · Zbl 0719.34002 |
| [13] | Lakshmikantham V , Bhaskar T G , Devi J V . Theory of Set Differential Equations in Metric Spaces. Cambridge: Cambridge Scientific Publisher, 2005 · Zbl 1153.34313 |
| [14] | Luo Y , Xie W Z . Existence of solution for impulsive differential inclusions with upper and lower solutions in the reverse order. Acta Math Sci, 2019, 39A (5): 1055- 1063 · Zbl 1449.34058 · doi:10.3969/j.issn.1003-3998.2019.05.008 |
| [15] | Mil’man V D , Myshkis A D . On the stability of motion in the presence of impulses. Siberian Math J, 1960, 1, 233- 237 · Zbl 1358.34022 |
| [16] | Plotnikova N . Averaging of impulsive differential inclusions. Mat Stud, 2005, 23 (1): 52- 56 · Zbl 1072.34046 |
| [17] | Plotnikov V A , Ivanov R P , Kitanov N M . Method of averaging for impulsive differential inclusions. Pliska Stud Math Bulg, 1998, 12, 191- 200 · Zbl 0946.49030 |
| [18] | Plotnikov V A , Komleva T . Averaging of set integro-differential equations. Applied Mathematics, 2011, 1 (2): 99- 105 |
| [19] | Plotnikov V A , Plotnikova L I . Averaging of differential inclusions with many-valued pulses. Ukrainian Mathematical Journal, 1995, 47 (11): 1741- 1749 · Zbl 0937.34011 · doi:10.1007/BF01057922 |
| [20] | Perestyuk N A , Plotnikov V A , Samoilenko A M , Skripnik N V . Differential Equations with Impulse Effects: Multivalued Right-Hand Sides with Discontinuities. Berlin: De Gruyter, 2011 · Zbl 1234.34002 |
| [21] | Plotnikov V A , Rashkov P I . Averaging in differential equations with Hukuhara derivative and delay. Functional Differential Equations, 2001, 3 (4): 371- 381 · Zbl 1046.34089 |
| [22] | Perestyuk N A , Skripnik N V . Averaging of set-valued impulsive systems. Ukrainian Mathematical Journal, 2013, 65 (1): 140- 157 · Zbl 1285.34039 · doi:10.1007/s11253-013-0770-1 |
| [23] | Samoilenko A M , Perestyuk N A . Impulsive Differential Equations. Singapore: World Scientific, 1995 · Zbl 0837.34003 |
| [24] | Skripnik N V . Averaging of impulsive differential inclusions with Hukuhara derivative. Nonlinear Oscil, 2007, 10 (3): 422- 438 · Zbl 1268.34039 · doi:10.1007/s11072-007-0033-x |
| [25] | Skrypnyk N V . On the averaging method for differential equations with Hukuhara’s derivative. Visn Cherniv Nats Univ, 2008, 374, 109- 115 · Zbl 1164.34410 |
| [26] | Skripnik N V . The full averaging of fuzzy impulsive differential inclusions. Surv Math Appl, 2010, 5, 247- 263 · Zbl 1413.34004 |
| [27] | Skripnik N V . The partial averaging of fuzzy impulsive differential inclusions. Different Integr Equat, 2011, 24 (7/8): 743- 758 · Zbl 1249.34131 |
| [28] | Skripnik N V . Averaging of impulsive differential inclusions with fuzzy right-hand side when the average is absent. Asian-European Journal of Mathematics, 2015, 8 (4): 1550086 · Zbl 1344.34052 · doi:10.1142/S1793557115500862 |
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