On the existence of solutions for first-order nonlinear conformable impulsive dynamic equations on time scales

Svetlin G. Georgiev; Rachid Bahloul; Faiza Shujat; Svetlin G. Georgiev; Rachid Bahloul; Faiza Shujat

doi:10.3934/era.2026384

Electronic Research Archive

2026, Volume 34, Issue 12: 9118-9137. doi: 10.3934/era.2026384

Previous Article Next Article

Research article Special Issues

On the existence of solutions for first-order nonlinear conformable impulsive dynamic equations on time scales

1.
Department of Mathematics, Sorbonne University, Paris, France
2.
Department of Mathematics, Sultan Moulay Slimane University, Beni Mellal, Morocco
3.
Department of Mathematics, College of Science, Taibah University, Madinah, Saudi Arabia

Received: 13 August 2026 Revised: 21 September 2026 Accepted: 08 October 2026 Published: 27 October 2026

Using a fixed point theorem due to Krasnosel'skii, we showed that the first-order nonlinear conformable impulsive dynamic equations on time scales (3.1) has a solution. Using the idea of contraction, we demonstrated that the solution is unique under a somewhat stricter inequality.
- conformable derivative,
- time scales,
- impulsive dynamic equations
Citation: Svetlin G. Georgiev, Rachid Bahloul, Faiza Shujat. On the existence of solutions for first-order nonlinear conformable impulsive dynamic equations on time scales[J]. Electronic Research Archive, 2026, 34(12): 9118-9137. doi: 10.3934/era.2026384

Related Papers:

Abstract

Using a fixed point theorem due to Krasnosel'skii, we showed that the first-order nonlinear conformable impulsive dynamic equations on time scales (3.1) has a solution. Using the idea of contraction, we demonstrated that the solution is unique under a somewhat stricter inequality.

References

[1]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
[2]	R. Khalil, M. Al. Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
[3]	W. Chung, Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math., 290 (2015), 150–158. https://doi.org/10.1016/j.cam.2015.04.049 doi: 10.1016/j.cam.2015.04.049
[4]	M. Bohner, V. F. Hatipoǧlu, Dynamic cobweb models with conformable fractional derivatives, Nonlinear Anal. Hybrid Syst., 32 (2019), 157–167. https://doi.org/10.1016/j.nahs.2018.09.004 doi: 10.1016/j.nahs.2018.09.004
[5]	T. Abdeljawad, Q. M. Al-Mdallal, F. Jarad, Fractional logistic models in the frame of fractional operators generated by conformable derivatives, Chaos, Solitons Fractals, 119 (2019), 94–101. https://doi.org/10.1016/j.chaos.2018.12.015 doi: 10.1016/j.chaos.2018.12.015
[6]	J. Rosales-García, J. A. Andrade-Lucio, O. Shulika, Conformable derivative applied to experimental Newton's law of cooling, Rev. Mex. Fis., 66 (2020), 224–227. https://doi.org/10.31349/revmexfis.66.224 doi: 10.31349/revmexfis.66.224
[7]	K. Alper, Explicit exact solutions to some one-dimensional conformable time fractional equations, Waves Random Complex Medium, 29 (2019), 124–137. https://doi.org/10.1080/17455030.2017.1416702 doi: 10.1080/17455030.2017.1416702
[8]	O. $\ddot{O}$zkan, A. Kurt, Exact solutions of fractional partial differential equation systems with conformable derivative, Filomat, 33 (2019), 1313–1322. https://doi.org/10.2298/FIL1905313O doi: 10.2298/FIL1905313O
[9]	M. Z. Saríkaya, F. Usta, On comparison theorems for conformable fractional differential equations, Int. J. Anal. Appl., 12 (2016), 207–214.
[10]	K. Tadeusz, Analysis of positive linear continuous–time systems using the conformable derivative, Int. J. Appl. Math. Comput. Sci., 28 (2018), 335–340. https://doi.org/10.2478/amcs-2018-0024 doi: 10.2478/amcs-2018-0024
[11]	F. Usta, A conformable calculus of radial basis functions and its applications, Int. J. Optim. Control, 8 (2018), 176–182. https://doi.org/10.11121/ijocta.01.2018.00544 doi: 10.11121/ijocta.01.2018.00544
[12]	F. Usta, M. Z. Saríkaya, The analytical solution of Van der Pol and Lienard differential equations within conformable fractional operator by retarded integral inequalities, Demonstr. Math., 52 (2019), 204–212. https://doi.org/10.1515/dema-2019-0017 doi: 10.1515/dema-2019-0017
[13]	B. Ahmad, M. Alghanmi, A. Alsaedi, R. P. Agarwal, On an impulsive hybrid system of conformable fractional differential equations with boundary conditions, Int. J. Syst. Sci., 51 (2020), 958–970. https://doi.org/10.1080/00207721.2020.1746437 doi: 10.1080/00207721.2020.1746437
[14]	G. E. Chatzarakis, K. Logaarasi, T. Raja, V. Sadhasivam, Interval oscillation criteria for impulsive conformable partial differential equations, Appl. Anal. Discrete Math., 13 (2019), 325–345.
[15]	E. Hernández, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Am. Math. Soc., 141 (2013), 1641–1649. https://doi.org/10.1090/S0002-9939-2012-11613-2 doi: 10.1090/S0002-9939-2012-11613-2
[16]	S. Liu, J. Wang, Y. Zhou, Optimal control of noninstantaneous impulsive differential equations, J. Franklin Inst., 354 (2017), 7668–7698. https://doi.org/10.1016/j.jfranklin.2017.09.010 doi: 10.1016/j.jfranklin.2017.09.010
[17]	A. Meraj, D. N. Pandey, Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions, Indian J. Pure Appl. Math., 51 (2020), 501–518. https://doi.org/10.1007/s13226-020-0413-9 doi: 10.1007/s13226-020-0413-9
[18]	B. Sundaravadivoo, Controllability analysis of nonlinear fractional order differential systems with state delay and non-instantaneous impulsive effects, Discrete Contin. Dyn. Syst. - Ser. S, 13 (2020), 2561–2573. https://doi.org/10.3934/dcdss.2020138 doi: 10.3934/dcdss.2020138
[19]	T. Liu, R. Xue, M. Barfeie, A unified framework for high-order compact finite differences using infinitely- and piecewise-smooth RBFs with polynomials, Calcolo, 63 (2026), 18. https://doi.org/10.1007/s10092-026-00684-1 doi: 10.1007/s10092-026-00684-1
[20]	J. Wang, M. Fečkan, A general class of impulsive evolution equations, Topol. Methods Nonlinear Anal., 46 (2015), 915–933. https://doi.org/10.12775/TMNA.2015.072 doi: 10.12775/TMNA.2015.072
[21]	J. Wang, M. Fečkan, Y. Tian, Stability analysis for a general class of non-instantaneous impulsive differential equations, Mediterr. J. Math., 14 (2017), 46. https://doi.org/10.1007/s00009-017-0867-0 doi: 10.1007/s00009-017-0867-0
[22]	D. Yang, J. Wang, D. O'Regan, On the orbital Hausdorff dependence of differential equations with non-instantaneous impulses, C. R. Math., 356 (2018), 150–171. https://doi.org/10.1016/j.crma.2018.01.001 doi: 10.1016/j.crma.2018.01.001
[23]	J. Andres, Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincare operators, Topol. Appl., 255 (2019), 126–140. https://doi.org/10.1016/j.topol.2019.01.008 doi: 10.1016/j.topol.2019.01.008
[24]	V. Slyn'ko, C. Tunc, Stability of abstract linear switched impulsive differential equations, Automatica, 107 (2019), 433–441. https://doi.org/10.1016/j.automatica.2019.06.001 doi: 10.1016/j.automatica.2019.06.001
[25]	J. Wang, M. Fečkan, Y. Zhou, A survey on impulsive fractional differential equations, Fract. Calc. Appl. Anal., 19 (2016), 806–831. https://doi.org/10.1515/fca-2016-0044 doi: 10.1515/fca-2016-0044
[26]	J. Wang, M. Fečkan, Y. Zhou, Fractional order differential switched systems with coupled nonlocal initial and impulsive conditions, Bull. Sci. Math., 141 (2017), 727–746. https://doi.org/10.1016/j.bulsci.2017.07.007 doi: 10.1016/j.bulsci.2017.07.007
[27]	W. Qiu, J. Wang, D. O'Regan, Existence and Ulam stability of solutions for conformable impulsive differential equations, Bull. Iran. Math. Soc., 46 (2020), 1613–1637. https://doi.org/10.1007/s41980-019-00347-8 doi: 10.1007/s41980-019-00347-8
[28]	M. Fečkan, J. Wang, Periodic impulsive fractional differential equations, Adv. Nonlinear Anal., 8 (2019), 482–496. https://doi.org/10.1515/anona-2017-0015 doi: 10.1515/anona-2017-0015
[29]	M. Li, J. Wang, D. O'Regan, Existence and Ulam's stability for conformable fractional differential equations with constant coefficients, Bull. Malays. Math. Sci. Soc., 42 (2019), 1791–1812. https://doi.org/10.1007/s40840-017-0576-7 doi: 10.1007/s40840-017-0576-7
[30]	J. Wang, M. Fečkan, Y. Zhou, Ulam's type stability of impulsive ordinary differential equations, J. Math. Anal. Appl., 395 (2012), 258–264. https://doi.org/10.1016/j.jmaa.2012.05.040 doi: 10.1016/j.jmaa.2012.05.040
[31]	E. R. Kaufmann, N. Kosmatov, Y. N. Raffoul, Impulsive dynamic equations on a time scale, Electron. J. Differ. Equations, 2008 (2008), 1–9.
[32]	B. Bayour, A. Hammoudi, D. F. M. Torres, A truly conformable calculus on time scales, preprint, arXiv: 1705.08928.
[33]	N. Benkhettoua, S. Hassania, D. F. M. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci., 28 (2016), 93–98. https://doi.org/10.1016/j.jksus.2015.05.003 doi: 10.1016/j.jksus.2015.05.003
[34]	S. Girel, Q. Richard, On the global asymptotic stability of an infection-age-structured competitive model, Math. Methods Appl. Sci., 2026 (2026), 1–33. https://doi.org/10.1002/mma.70967 doi: 10.1002/mma.70967
[35]	T. Liu, R. Xue, A convergent multi-step efficient iteration method to solve nonlinear equation systems, J. Appl. Math. Comput., 71 (2024), 2571–2588. https://doi.org/10.1007/s12190-024-02324-9 doi: 10.1007/s12190-024-02324-9
[36]	S. G. Georgiev, R. Bahloul, Some results of generalization bilateral Laplace transform, Gulf J. Math., 22 (2026), 1–21. https://doi.org/10.56947/gjom.v22i1.3901 doi: 10.56947/gjom.v22i1.3901
[37]	J. Henderson, A. Ouahab, S. Youcefi, Existence results for phi-Laplacian impulsive differential equations with periodic conditions, AIMS Math., 4 (2019), 1610–1633. https://doi.org/10.3934/math.2019.6.1610 doi: 10.3934/math.2019.6.1610

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)