Ulam’s stability of impulsive sequential coupled system of mixed order derivatives

Document Type : Research Paper


Department of Mathematics, University of Peshawar, 25000, Pakistan


This manuscript is devoted to establishing Hyers–Ulam stability for a class of non-linear impulsive coupled sequential fractional differential equations with multi point boundary conditions on a closed interval [0,T] with Caputo fractional derivative having non-instantaneous impulses. Sufficient conditions are introduced that guarantee the existence of a unique solution to the proposed problem. Furthermore, Hyers–Ulam stability of the proposed model is also presented and an example is provided to authenticate the theoretical results.


[1] N. Ahmad, Z. Ali, K. Shah, A. Zada and G. Rahman, Analysis of implicit type nonlinear dynamical problem of
impulsive fractional differential equations, Complexity 2018 (2018) 1–15.
[2] B. Ahmad and S. Sivasundaram, On four point nonlocal boundary value problems of nonlinear integro-differential
equations of fractional order, Appl. Math. Comput. 217 (2010) 480–487.
[3] Z. Ali, A. Zada and K. Shah, On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential
equations, Bull. Malays. Math. Sci. Soc. 42(5) (2019) 2681–2699.
[4] M. Altman, A fixed point theorem for completely continuous operators in banach spaces, Bull. Acad. polon. Sci.
3 (1955) 409–413.
[5] M. El-Shahed and Nieto, Non trivial solutions for a nonlinear multi-point boundary value problem of fractional
order, Comput. Math. Appl. 59 (2010) 3438–3443.
[6] S. M. Jung, Hyers–Ulam stability of linear differential equations of first order, Appl. Math. Lett. 19 (2006)
[7] R. A. Khan and K. Shah, Existence and uniqueness of solutions to fractional order multi-point boundary value
problems, Commun. Appl. Anal. 19 (2015) 515–526.
[8] A. Khan, M. I. Syam, A. Zada and H. Khan, Stability analysis of nonlinear fractional differential equations with
Caputo and Riemann-Liouville derivatives, Eur. Phys. J. Plus 133(264) (2018) 1–9.
[9] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Application of Fractional Differential Equation,
North-Holl and Mathematics Studies, 204. Elsevier Science B. V, Amsterdam (2006).
[10] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equation, Wiley,
New York (1993).
[11] M. Obloza, Hyers stability of the linear differential equation, Rocznik NaukDydakt, Prace Mat. 13 (1993) 259–270.
[12] K. B. Oldham, Fractional differential equations in electrochemistry, Advances in Engineering software. 41 (2010)
[13] U. Riaz, A. Zada, Z. Ali, M. Ahmad, J. Xu and Z. Fu Analysis of nonlinear coupled systems of impulsive fractional
differential equations with Hadamard derivatives, Math. Probl. Eng. 2019 (2019) 1–20.
[14] U. Riaz, A. Zada, Z. Ali, Y. Cui and J. Xu, Analysis of coupled systems of implicit impulsive fractional differential
equations involving Hadamard derivatives, Adv. Difference Equ. 2019(226) (2019) 1–27.
[15] F. A. Rihan, Numerical Modeling of Fractional Order Biological Systems, Abs. Appl. Anal. (2013).
[16] R. Rizwan, A. Zada and X. Wang, Stability analysis of non linear implicit fractional Langevin equation with
noninstantaneous impulses, Adv. Difference Equ. 2019(85) (2019) 1–31.
[17] J. Sabatier, O. P. Agrawal and J. A. T. Machado, Advances in Fractional Calculus, Dordrecht: Springer. (2007).[18] K. Shah and R. A. Khan, Existence and uniqueness of positive solutions to a coupled system of nonlinear fractional
order differential equations with anti-periodic boundary conditions, Differ. Equ. Appl. 7(2) (2015) 245–262.
[19] V. E. Tarasov, Fractional Dynamics: Application of Fractional Calculus to Dynamics of particles, Fields and
Media,Springer, Heidelberg; Higher Education Press, Beijing. (2010).
[20] S. M. Ulam, A Collection of the Mathematical Problems, Interscience, New York. (1960).
[21] C. Urs, Coupled fixed point theorem and application to periodic boundary value problem, Miskol. Math. Notes 14
(2013) 323–333.
[22] B. M. Vintagre, I. Podlybni, A. Hernandez and V. Feliu, Some approximations of fractional order operators used
in control theory and applications, Fract. Calc. Appl. Anal. 3(3) (2000) 231–248.
[23] J. Wang, L. Lv and W. Zhou, Ulam stability and data dependence for fractional differential equations with Caputo
derivative, Electron. J. Qual. Theory Differ. Equ. 63 (2011) 1–10.
[24] J. Wang, K. Shah and A. Ali, Existence and Hyers–Ulam stability of fractional non-linear impulsive switched
coupled evolution equation, Math. Meth. Appl. Sci. 41(6) (2018) 2392–2402.
[25] J. Wang, A. Zada and H. Waheed, Stability analysis of a coupled system of nonlinear implicit fractional antiperiodic boundary value problem, Math. Meth. App. Sci. 42(18) (2019) 6706–6732.
[26] J. Wang and Y. Zhang, On the concept and existence of solutions for fractional impulsive systems with Hadamard
derivatives, Appl. Math. Lett. 39 (2014) 85–90.
[27] A. Zada and S. Ali, Stability Analysis of multi-point boundary value problem for sequential fractional differential
equations with noninstantaneous impulses, Int. J. Nonlinear Sci. Numer. Simul. 19 (7) (2018) 763–774.
[28] A. Zada and S. Ali, Stability of integral Caputo–type boundary value problem with noninstantaneous impulses,
Int. J. Appl. Comput. Math. 2019(55) (2019) 1–18.
[29] A. Zada, W. Ali and S. Farina, Hyers–Ulam stability of nonlinear differential equations with fractional integrable
impulses, Math. Meth. App. Sci. 40(15) (2017) 5502–5514.
[30] A. Zada, S. Ali and Y. Li, Ulam–type stability for a class of implicit fractional differential equations with non–
instantaneous integral impulses and boundary condition, Adv. Difference Equ. 2017(317) (2017) 1–26.
[31] A. Zada, W. Ali and C. Park, Ulam’s type stability of higher order nonlinear delay differential equations via
integral inequality of Gro¨nwall–Bellman–Bihari’s type, Appl. Math. Comput. 350 (2019) 60–65.
[32] A. Zada, S. Faisal and Y. Li, On the Hyers–Ulam stability of first order impulsive delay differential equations, J.
Funct. Spaces 2016 (2016) 1–6.
[33] A. Zada, F. U. Khan, U. Riaz and T. Li, Hyers–Ulam stability of linear summation equations, Punjab Univ. j.
math. 49(1) (2017) 19–24.
[34] A. Zada and U. Riaz, Kallman–Rota type inequality for discrete evolution families of bounded linear operators,
Fract. Differ. Calc. 7(2) (2017) 311–324.
[35] A. Zada, U. Riaz and F. Khan, Hyers–Ulam stability of impulsive integral equations, Boll. Unione Mat. Ital. 12(3)
(2019) 453–467.
Volume 13, Issue 1
March 2022
Pages 57-73
  • Receive Date: 13 April 2020
  • Revise Date: 05 May 2020
  • Accept Date: 08 May 2020
  • First Publish Date: 07 September 2021