Abstract
In this paper, we investigate the orbital Hausdorff continuous dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses. The concepts of orbital Hausdorff continuous dependence and stability are used to characterize the relations of solution corresponding to the impulsive moments and the difference between the impulsive moments and the junction points in the sense of the Hausdorff distance. Then, we establish sufficient conditions to guarantee the orbital Hausdorff continuous dependence and stability on their respective trajectories. Finally, two examples are given to illustrate our theoretical results.
(Communicated by Michal Fečkan)
Acknowledgement
The authors want to thank the reviewers for spending their precious time to read this manuscript and insightful comments that helped improve the presentation of this article.
References
[1] Andres, J.—Ludvík, P.: Parametric topological entropy for multivalued maps and differential inclusions with nonautonomous impulses, Internat. J. Bifur. Chaos 33 (2023), Art. ID 2350113.10.1142/S0218127423501134Search in Google Scholar
[2] Bainov, D.—Simeonov, P.: Theory of Impulsive Differential Equations. Advances in Mathematics for Applied Sciences, World Scientific, Singapore, 1995.10.1142/9789812831804Search in Google Scholar
[3] Bainov, D. D.—Milusheva, S. D.: Investigation of partially-multiplicative averaging for a class of functional differential equations with variable structure and impulses, Tsukuba J. Math. 9 (1981), 1–19.10.21099/tkbjm/1496160190Search in Google Scholar
[4] Bainov, D. D.—Milusheva, S. D.: Justification of the averaging method for a system of functional-differential equations with variable structure and impulses, Appl. Math. Optim. 16 (1987), 19–36.10.1007/BF01442183Search in Google Scholar
[5] Chen, Y.—Wang, J.: Continuous dependence of solutions of integer and fractional order non-instantaneous impulsive equations with random impulsive and junction points, Mathematics 7 (2019), Art. No. 331.10.3390/math7040331Search in Google Scholar
[6] Chiu, K.—Sepúlveda, I.: Nonautonomous impulsive differential equations of alternately advanced and retarded type, Filomat 37 (2023), 7813–7829.10.2298/FIL2323813CSearch in Google Scholar
[7] Ding, Y.—O’Regan, D.—Wang, J.: Stability analysis for conformable non-instantaneous impulsive differential equations, Bull. Iranian Math. Soc. 48 (2022), 1435–1459.10.1007/s41980-021-00595-7Search in Google Scholar
[8] Dishliev, A. B.—Bainov, D. D.: Continuous dependence on the initial condition of the solution of asystem of differential equations with variable structure and with impulses, Publ. Res. Inst. Math. Sci. 23 (1987), 923–936.10.2977/prims/1195175864Search in Google Scholar
[9] Dishliev, A. B.—Bainov, D. D.: Dependence upon initial conditions and parameter of solutions of impulsive differential equations with variable structure, Internat. J. Theoret. Phys. 29 (1990), 655–675.10.1007/BF00672039Search in Google Scholar
[10] Dishlieva, K. G.—Dishliev, A. B.—Petkova, S. A.: Death of the solutions of systems differential equations with variable structure and impulses, Internat. J. Differ. Equ. Appl. 29 (2012), 169–181.Search in Google Scholar
[11] Dishlieva, K. G.—Dishliev, A. B.: Uniformly finally bounded solutions to systems of differential equations with variable structure and impulses, Electron. J. Differ. Equ. 2014 (2014), 1–9.10.1186/1687-1847-2014-1Search in Google Scholar
[12] Dishlieva, K. G.: Periodic solutions of systems of autonomous differential equations with variable structure and impulses, Internat. J. Pure Appl. Math. 105 (2015), 599–625.Search in Google Scholar
[13] Dishlieva, K.—Antonov, A.: Hausdorff Metric and Differential Equations with Variable Structure and Impulses, Technical University of Sofia, Bulgaria, 2015.10.12732/ijpam.v105i4.3Search in Google Scholar
[14] Fan, Z.—Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions, J. Funct. Anal. 258 (2010), 1709–1727.10.1016/j.jfa.2009.10.023Search in Google Scholar
[15] Fernandez, A.—Ali, S.—Zada, A.: On non-instantaneous impulsive fractional differential equations and their equivalent integral equations, Math. Methods Appl. Sci. 44 (2021), 13979–13988.10.1002/mma.7669Search in Google Scholar
[16] Hernández, E.: Abstract impulsive differential equations without predefined time impulses, J. Math. Anal. Appl. 491 (2020), Art. ID 124288.10.1016/j.jmaa.2020.124288Search in Google Scholar
[17] Hernández, E.—O’Regan, D.: On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc. 141 (2013), 1641–1649.10.1090/S0002-9939-2012-11613-2Search in Google Scholar
[18] Hernández, E.—Pierri, M.—O’Regan, D.: On abstract differential equations with non instantaneous impulses, Topol. Methods Nonlinear Anal. 46 (2015), 1067–1085.Search in Google Scholar
[19] Ji, D.—Yang, L.—Zhang, J.: Almost periodic functions on Hausdorff almost periodic time scales, Adv. Differ. Equ. 2017 (2017), 103.10.1186/s13662-017-1159-5Search in Google Scholar
[20] Li, C.—Hui, F.—Li, F.: Stability of differential systems with impulsive effects, Math. 11 (2023), Art. 4382.10.3390/math11204382Search in Google Scholar
[21] Liu, S.—Debbouche, A.—Wang, J.: On the iterative learning control for stochastic impulsive differential equations with randomly varying trial lengths, J. Comput. Appl. Math. 312 (2017), 47–57.10.1016/j.cam.2015.10.028Search in Google Scholar
[22] Lu, W.—Pinto, M.—Xia, Y.: Smooth stable manifolds for the non-instantaneous impulsive equations with applications to Duffing oscillators, Proc. R. Soc. A Math. Phys. Eng. Sci. 478 (2022), 2259.10.1098/rspa.2021.0957Search in Google Scholar PubMed PubMed Central
[23] Mil’man, V. D.—Myshkis, A. D.: On stability of motion in the presence of impulses, Sibirskii Math. J. 1 (1960), 233–237.Search in Google Scholar
[24] Sendov, B.: Hausdorff Approximations, Springer Science and Business Media, 1990.10.1007/978-94-009-0673-0Search in Google Scholar
[25] Stamova, I.—Stamov, G.: Applied Impulsive Mathematical Models, Springer, 2016.10.1007/978-3-319-28061-5Search in Google Scholar
[26] Sun, J.—Chu, J.—Chen, H.: Periodic solution generated by impulses for singular differential equation, J. Math. Anal. Appl. 404 (2013), 562–569.10.1016/j.jmaa.2013.03.036Search in Google Scholar
[27] Wang, J.: Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett. 73 (2017), 157–162.Search in Google Scholar
[28] Wang, J.—Fec̆kan, M.—Debbouche, A.: Time optimal control of a system governed by non-instantaneous impulsive differential equations, J. Optim. Theory Appl. 182 (2019), 573–587.10.1007/s10957-018-1313-6Search in Google Scholar
[29] Wang, J.—Fec̆kan, M.—Zhou, Y.: Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bull. Sci. Math. 141 (2017), 727–746.10.1016/j.bulsci.2017.07.007Search in Google Scholar
[30] Wang, J.: Stability of noninstantaneous impulsive evolution equations, Appl. Math. Lett. 73 (2017), 157–162.10.1016/j.aml.2017.04.010Search in Google Scholar
[31] Yang, T.: Impulsive Control Theory, Springer, Berlin, 2001.Search in Google Scholar
[32] Yang, P.—Wang, J.—Fec̆kan, M.: Periodic nonautonomous differential equations with noninstantaneous impulsive effects, Math. Methods Appl. Sci. 42 (2019), 3700–3720.10.1002/mma.5606Search in Google Scholar
[33] Yang, D.—Wang, J.—O’Regan, D.: Asymptotic properties of the solutions of nonlinear non-instantaneous impulsive differential equations, J. Franklin Inst. 354 (2017), 6978–7011.10.1016/j.jfranklin.2017.08.011Search in Google Scholar
[34] Yang, D.—Wang, J.—O’Regan, D.: On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses, C. R. Math. 356 (2018), 150–171.10.1016/j.crma.2018.01.001Search in Google Scholar
[35] Zhang, G.—Song, M.—Liu, M.: Exponential stability of the exact solutions and the numerical solutions for a class of linear impulsive delay differential equations, J. Comput. Appl. Math. 285 (2015), 32–44.10.1016/j.cam.2015.01.034Search in Google Scholar
[36] Zhang, D.—He, F.—Han, S.: An efficient approach to directly compute the exact Hausdorff distance for 3D point sets, Integrated Comput. Aided Eng. 24 (2017), 261–277.10.3233/ICA-170544Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Generalized Sasaki mappings in d0-Algebras
- On a theorem of Nathanson on Diophantine approximation
- Constructing infinite families of number fields with given indices from quintinomials
- Partitions into two Lehmer numbers in ℤq
- Fundamental systems of solutions of some linear differential equations of higher order
- On k-Circulant matrices involving the Lucas numbers of even index
- Explicit formulae for the Drazin inverse of the sum of two matrices
- On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
- Novel generalized tempered fractional integral inequalities for convexity property and applications
- Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
- The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
- Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
- Improvements on the Leighton oscillation theorem for second-order dynamic equations
- Topogenous orders on forms
- Comparison of topologies on fundamental groups with subgroup topology viewpoint
- An elementary proof of the generalized Itô formula
- Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951
Articles in the same Issue
- Generalized Sasaki mappings in d0-Algebras
- On a theorem of Nathanson on Diophantine approximation
- Constructing infinite families of number fields with given indices from quintinomials
- Partitions into two Lehmer numbers in ℤq
- Fundamental systems of solutions of some linear differential equations of higher order
- On k-Circulant matrices involving the Lucas numbers of even index
- Explicit formulae for the Drazin inverse of the sum of two matrices
- On nonoscillation of fractional order functional differential equations with forcing term and distributed delays
- Novel generalized tempered fractional integral inequalities for convexity property and applications
- Convergence of α-Bernstein-Durrmeyer operators about a collection of measures
- The problem of finding eigenvalues and eigenfunctions of boundary value problems for an equation of mixed type
- Orbital Hausdorff dependence and stability of the solution to differential equations with variable structure and non-instantaneous impulses
- Improvements on the Leighton oscillation theorem for second-order dynamic equations
- Topogenous orders on forms
- Comparison of topologies on fundamental groups with subgroup topology viewpoint
- An elementary proof of the generalized Itô formula
- Asymptotic normality for kernel-based test of conditional mean independence in Hilbert space
- Advancing reliability and medical data analysis through novel statistical distribution exploration
- Historical notes on the 75th volume of Mathematica Slovaca - Authors of the first issue from 1951