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Stability Analysis for Conformable Non-instantaneous Impulsive Differential Equations

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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.

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References

  1. Milman, V.D., Myshkis, A.D.: On the stability of motion in the presence of impulses. Sibirskii Matematicheskii Zhurnal 1, 233–237 (1960)

    MathSciNet  Google Scholar 

  2. 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)

    MathSciNet  MATH  Google Scholar 

  3. Henderson, J., Ouahab, A.: Impulsive differential inclusions with fractional order. Comput. Math. Appl. 59, 1191–1226 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Bainov, D.D., Simenov, P.S.: Systems with Impulse Effect: Stability Theory and Applications. Ellis Horwood, Amsterdam (1989)

    Google Scholar 

  5. Nenov, S.I.: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. Theory Methods Appl. 36, 881–890 (1999)

    MathSciNet  MATH  Google Scholar 

  6. 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)

    MathSciNet  MATH  Google Scholar 

  7. Fan, Z., Li, G.: Existence results for semilinear differential equations with nonlocal and impulsive conditions. J. Funct. Anal. 258, 1709–1727 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Hernández, E., O’Regan, D.: On a new class of abstract impulsive differential equations. Proc. Am. Math. Soc. 141, 1641–1649 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Wang, J., Fečkan, M.: A general class of impulsive evolution equations. Topol. Methods Nonlinear Anal. 46, 915–933 (2015)

    MathSciNet  MATH  Google Scholar 

  10. 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)

    Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. 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)

    Google Scholar 

  13. 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)

    MathSciNet  MATH  Google Scholar 

  14. 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)

    MathSciNet  MATH  Google Scholar 

  15. Khalil, R., Horani, M.. Al., Yousef, A., Sababheh, M.: A new definition of fractional derivative. J. Comput. Appl. Math. 264, 65–70 (2014)

    MathSciNet  MATH  Google Scholar 

  16. 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)

    Google Scholar 

  17. Arqub, O.A., Al-Smadi, M.: Fuzzy conformable fractional differential equations: novel extended approach and new numerical solutions. Soft. Comput. 24, 12501–12522 (2020)

    MATH  Google Scholar 

  18. 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)

    MathSciNet  MATH  Google Scholar 

  19. 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)

    Google Scholar 

  20. Akdemir, A.O., Ekinci, A., Set, E.: Conformable fractional integrals and related new integral inequalities. J. Nonlinear Convex Anal. 18, 661–674 (2017)

    MathSciNet  MATH  Google Scholar 

  21. 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)

    Google Scholar 

  22. 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)

    Google Scholar 

  23. 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)

    MathSciNet  Google Scholar 

  24. 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)

    Google Scholar 

  25. 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)

    MathSciNet  MATH  Google Scholar 

  26. 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

    MATH  Google Scholar 

  27. 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)

    MathSciNet  MATH  Google Scholar 

  28. Ding, Y., Fečkan, M., Wang, J.: Conformable linear and nonlinear non-instantaneous impulsive differential equations. Electron. J. Differ. Equ. 2020, 1–19 (2020)

    Google Scholar 

  29. Abdeljawad, T.: On conformable fractional calculus. J. Comput. Appl. Math. 279, 57–66 (2015)

    MathSciNet  MATH  Google Scholar 

  30. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York (1970)

    MATH  Google Scholar 

  31. Samoilenko, A.M., Perestyuk, N.A., Chapovsky, Y.: Impulsive Differential Equations. World Scientific, Singapore (1995)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou, 550025, People’s Republic of China

    Yuanlin Ding & JinRong Wang

  2. School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

    Donal O’Regan

  3. School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, Shandong, People’s Republic of China

    JinRong Wang

Authors
  1. Yuanlin Ding
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  2. Donal O’Regan
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  3. JinRong Wang
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Correspondence to JinRong Wang.

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Communicated by Majid Gazor.

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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).

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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

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  • DOI: https://doi.org/10.1007/s41980-021-00595-7

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