Abstract
In this article, we analyze a class of conformable non-instantaneous impulsive differential equations and obtain their solutions using the non-instantaneous impulsive Cauchy matrix. We derive a suitable formula for the solution of conformable nonhomogeneous linear non-instantaneous impulsive perturbed problems and we study its exponential stability. We also investigate nonlinear non-instantaneous impulsive equations and provide some conditions needed to establish existence and uniqueness of their solutions and then we present a result which guarantees Ulam–Hyers–Rassias stability. Finally, an example is given to illustrate the theory.
Similar content being viewed by others
References
Milman, V.D., Myshkis, A.D.: On the stability of motion in the presence of impulses. Sibirskii Matematicheskii Zhurnal 1, 233–237 (1960)
Wang, J., Fečkan, M., Zhou, Y.: On the new concept of solutions and existence results for impulsive fractional evolution equations. Dyn. Partial Differ. Equ. 8, 345–361 (2011)
Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)
Bainov, D.D., Simenov, P.S.: Systems with Impulse Effect: Stability Theory and Applications. Ellis Horwood, Amsterdam (1989)
Nenov, S.I.: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. Theory Methods Appl. 36, 881–890 (1999)
Akhmet, M.U., Alzabut, J., Zafer, A.: Perron’s theorem for linear impulsive differential equations with distributed delay. J. Comput. Appl. Math. 193, 204–218 (2006)
Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258, 1709–1727 (2010)
Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)
Wang, J., Fečkan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46, 915–933 (2015)
Malik, M., Kumar, A., Fečkan, M.: Existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. J. King Saud Univ. Sci. 30, 204–213 (2018)
Meraj, A., Pandey, D.N.: Approximate controllability of non-autonomous Sobolev type integro-differential equations having nonlocal and non-instantaneous impulsive conditions. Indian J. Pure Appl. Math. 51, 501–518 (2020)
Agarwal, R., Shristova, S., O’Regan, D.: Ulam type stability results for non-instantaneous impulsive differential equations with finite state dependent delay. Dyn. Syst. Appl. 28, 47–61 (2018)
Kumar, A., Muslim, M., Sakthivel, R.: Controllability of the second-order nonlinear differential equations with non-instantaneous impulses. J. Dyn. Control Syst. 24, 325–342 (2018)
Li, M., Wang, J., O’Regan, D.: Positive almost periodic solution for a noninstantaneous impulsive Lasota–Wazewska model. Bull. Iran. Math. Soc. 46, 851–864 (2020)
Khalil, R., Horani, M.. Al., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)
Nazir, A., Ahmed, N., Khan, U., Mohyud-Din, S.T., Nisar, K.S., Khan, I.: An advanced version of a conformable mathematical model of Ebola virus disease in Africa. Alex. Eng. J. 59, 3261–3268 (2020)
Arqub, O.A., Al-Smadi, M.: Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions. Soft. Comput. 24, 12501–12522 (2020)
Xiao, G., Wang, J., O’Regan, D.: Existence, uniqueness and continuous dependence of solutions to conformable stochastic differential equations. Chaos Solitons Fractals 139, 110269 (2020)
Al-Zhour, Z., Al-Mutairi, N., Alrawajeh, F., Alkhasawneh, R.: Series solutions for the Laguerre and Lane–Emden fractional differential equations in the sense of conformable fractional derivative. Alex. Eng. J. 58, 1413–1420 (2019)
Akdemir, A.O., Ekinci, A., Set, E.: Conformable fractional integrals and related new integral inequalities. J. Nonlinear Convex Anal. 18, 661–674 (2017)
Talafha, A.G., Alqaraleh, S.M., Al-Smadi, M., Hadid, S., Momani, S.: Analytic solutions for a modified fractional three wave interaction equations with conformable derivative by unified method. Alex. Eng. J. 59, 3731–3739 (2020)
Korpinar, Z., Tchier, F., Inc, M., Bousbahi, F., Tawfiq, F.M.O., Akinlar, M.A.: Applicability of time conformable derivative to Wick-fractional-stochastic PDEs. Alex. Eng. J. 59, 1485–1493 (2020)
Rosales-Garcia, J., Andrade-Lucio, J.A., Shulika, O.: Conformable derivative applied to experimental Newton’s law of cooling. Rev. Mex. Fis. 66, 224–227 (2020)
Al Qurashi, M.. M.: Conserved vectors with conformable derivative for certain systems of partial differential equations with physical applications. Open Phys. 18, 164–169 (2020)
Qiu, W., Wang, J., O’Regan, D.: Existence and Ulam stability of solutions for conformable impulsive differential equations. Bull. Iran. Math. Soc. 46, 1613–1637 (2020)
Qiu, W., Fečkan, M., O’Regan, D., Wang, J.: Convergence analysis for iterative learning control of conformable impulsive differential equations. Bull. Iran. Math. Soc. (2021). https://doi.org/10.1007/s41980-020-00510-6
Li, M., Wang, J., O’Regan, D.: Existence and Ulam’s stability for conformable fractional differential equations with constant coefficients. Bull. Malays. Math. Sci. Soc. 40, 1791–1812 (2019)
Ding, Y., Fečkan, M., Wang, J.: Conformable linear and nonlinear non-instantaneous impulsive differential equations. Electron. J. Differ. Equ. 2020, 1–19 (2020)
Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Samoilenko, A.M., Perestyuk, N.A., Chapovsky, Y.: Impulsive Differential Equations. World Scientific, Singapore (1995)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Majid Gazor.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012) and Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016).
Rights and permissions
About this article
Cite this article
Ding, Y., O’Regan, D. & Wang, J. Stability Analysis for Conformable Non-instantaneous Impulsive Differential Equations. Bull. Iran. Math. Soc. 48, 1435–1459 (2022). https://doi.org/10.1007/s41980-021-00595-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s41980-021-00595-7
Keywords
- Conformable derivative
- Non-instantaneous impulsive differential equations
- Exponential stability
- Ulam–Hyers–Rassias stability