Advertisement

Results in Mathematics

, 74:142 | Cite as

Mathematical Analysis of Implicit Impulsive Switched Coupled Evolution Equations

  • Asma
  • Ghaus ur RahmanEmail author
  • Kamal Shah
Article
  • 80 Downloads

Abstract

In the present manuscript a novel type of implicit switched coupled evolution system is studied. The underlying system is formulated with fractional order differential operator while incorporating impulses in the solution. Reducing the proposed model into fixed point problem, results for the existence and uniqueness of solution are exhibited. Also, we established results related to Hyers–Ulam type stability of the solution. An illustrative example is also solved to support the obtained theoretical results.

Keywords

Caputo fractional derivative switched couple system boundary conditions Hyers–Ulam stability fractional order differential equation 

Mathematics Subject Classification

34A08 34B15 34B27 

Notes

Acknowledgements

The authors are thankful to Higher Education Commission for approving SRGP project no: 21-1657/SRGP/R&D/HEC/2017.

Author Contributions

All the authors equally contributed and approved the final draft of the manuscript.

Funding

The research is supported by Higher Education Commission(HEC) Islamabad through Project No. (21-1657/SRGP/R&D/HEC/2017) and the present research work is a part of the aforementioned project.

Compliance with Ethical Standards

Conflict of interest

It is declared that no competing interest exists among the authors regarding this manuscript.

References

  1. 1.
    Agarwal, R.P., Asma, Lupulescu, V., O’Regan, D.: Fractional semilinear equations with causal operators. Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mate. 111(1), 257–269 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agarwal, R.P., Asma, Lupulescu, V., O’Regan, D.: \(\text{ L }^{{\rm p}} \)-solutions for a class of fractional integral equations. J. Integral Equ. Appl. 29, 251–270 (2017)CrossRefGoogle Scholar
  3. 3.
    Ali, A.: Ulam type stability analysis of implicit impulsive fractional differential equations. Dissertations Math. (2017)Google Scholar
  4. 4.
    Ali, Z., Zada, A., Shah, K.: On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations. Bull. Malays. Math. Sci. Soc. (2018).  https://doi.org/10.1007/s40840-018-0625-x CrossRefzbMATHGoogle Scholar
  5. 5.
    Ali, A., Rabieib, F., Shah, K.: On Ulam’s type stability for a class of impulsive fractional differential equations with nonlinear integral boundary conditions. J. Nonlinear Sci. Appl. 10, 4760–4775 (2017)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Ali, Z., Zada, A., Shah, K.: Existence and stability analysis of three point boundary value problem. Int. J. Appl. Comput. Math. (2017).  https://doi.org/10.1007/s40819-017-0375-8 MathSciNetCrossRefGoogle Scholar
  7. 7.
    Ahmad, N., Ali, Z., Shah, K., Zada, A., Rahman, Gu: Analysis of implicit type nonlinear dynamical problem of impulsive fractional differential equations. Complexity 2018, 6423974 (2018).  https://doi.org/10.1155/2018/6423974 CrossRefzbMATHGoogle Scholar
  8. 8.
    Brillouët-Belluot, N., Brzdȩk, J., Ciepliński, K.: On some recent developments in Ulam’s type stability. Abstr. Appl. Anal. 2012, 716936 (2012).  https://doi.org/10.1155/2012/716936 MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York (2006)CrossRefGoogle Scholar
  10. 10.
    Benchohra, B., Bouriah, S.: Existence and stability results for nonlinear boundary value problem for implicit differential equations of fractional order. Moroc. J. Pure. Appl. Anal. 1(1), 22–37 (2015)CrossRefGoogle Scholar
  11. 11.
    Benchohra, B., Bouriah, S., Henderson, J.: Existence and stability results for nonlinear implicit neutral fractional differential equations with finite delay and impulses. Commun. Appl. Nonlinear Anal. 22, 46–67 (2015)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Chiu, K.-S., Li, T.: Oscillatory and periodic solutions of differential equations with piecewise constant generalized mixed arguments. Math. Nachr. 2019, 1–2 (2019).  https://doi.org/10.1002/mana.201800053 CrossRefGoogle Scholar
  13. 13.
    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. U.S.A. 27, 222–224 (1941)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hyers, D.H., Isac, G., Rassias, T.M.: Stability of Functional Equations in Several Variables. Birkhäiuser, Boston (1998)CrossRefGoogle Scholar
  15. 15.
    Ibrahim, R.W.: Generalized Ulam-Hyers stability for fractional differential equations. Int. J. Math. 23, 1250056 (2012)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Jung, M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 19, 854–858 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Jung, S.M.: On the Hyers–Ulam stability of functional equations that have the quadratic property. J. Math. Appl. 222, 126–137 (1998)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Jiang, C., Zhang, F., Li, T.: Synchronization and antisynchronization of N-coupled fractional order complex chaotic systems with ring connection. Math. Methods Appl. Sci. 41, 2625–2638 (2018)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Khan, A., Shah, K., Li, Y., Khan, T.S.: Ulam type stability for a coupled systems of boundary value problems of nonlinear fractional differential equations. J. Funct. Spaces 2017, 3046013 (2017)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  21. 21.
    Komarova, N.L., Newell, A.C.: Nonlinear dynamics of sand banks and sand waves. J. Fluid Mech. 415, 285–321 (2000)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Li, T., Zada, T.: Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equ. 1, 1–8 (2016)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Li, T., Zada, T., Faisal, S.: Hyers–Ulam stability of nth order linear differential equations. J. Nonlinear Sci. Appl. 9, 2070–2075 (2016)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Michalski, M.W.: Derivatives of noninteger order and their applications, Dissertations Math, vol. 328 (1993)Google Scholar
  25. 25.
    Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)zbMATHGoogle Scholar
  26. 26.
    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)zbMATHGoogle Scholar
  27. 27.
    Riecke, H.: Self-trapping of traveling-wave pulses in binary mixture convection. Phys. Rev. Lett. 68, 301–304 (1992)CrossRefGoogle Scholar
  28. 28.
    Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Rassias, T.M.: On the stability of functional equations and a problem of Ulam. Acta Appl. Math. 62, 23–130 (2000)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shah, K., Khan, R.A.: Multiple positive solutions to a coupled systems of nonlinear fractional differential equations. SpringerPlus 5, 1–20 (2016)CrossRefGoogle Scholar
  31. 31.
    Shah, K., Khalil, H., Khan, R.A.: Investigation of positive solution to a coupled system of impulsive boundary value problems for nonlinear fractional order differential equations. Chaos Solitons Fractals 77, 240–246 (2015)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Shah, K., Tunc, C.: Existence theory and stability analysis to a system of boundary value problem. J. Taibah Univ. Sci. (2017).  https://doi.org/10.1016/j.jtusci.2017.06.002 CrossRefGoogle Scholar
  33. 33.
    Samoilenko, A.M., Perestyuk, N.A., Chapovsky, Y.: Impulsive Differential Equations. World Scientific, Singapore (1995)CrossRefGoogle Scholar
  34. 34.
    Tang, S., Zada, A., Faisal, S., El-Sheikh, M.M.A., Li, T.: Stability of higher order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 9, 4713–4721 (2016)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ulam, S.M.: A Collection of the Mathematical Problems. Interscience, New York (1960)zbMATHGoogle Scholar
  36. 36.
    Urs, C.: Coupled fixed point theorems and applications to periodic boundary value problems. Miskolc Math. Notes 14, 323–33 (2013)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Wang, J., Lv, L., Zhou, W.: Ulam stability and data dependence for fractional differential equations with Caputo derivative. Electron. J. Qual. Theory Differ. Equ. 63, 1–10 (2011)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Wang, P., Li, C., Zhang, J., Li, T.: Quasilinearization method for first-order impulsive integrodifferential equations. Electron. J. Diff. Equ. 2019, 1–14 (2019)CrossRefGoogle Scholar
  39. 39.
    Zada, A., Shaleena, S., Li, T.: Stability analysis of higher order nonlinear differential equations in \(\beta \)-normed spaces. Math. Methods Appl. Sci. 42, 1151–1166 (2019)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Zada, A., Ali, W., Farina, S.: Hyers–Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods Appl. Sci. (2017).  https://doi.org/10.1002/mma.4405 MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Zada, A., Wang, P., Lassoued, D., Li, T.: Connections between Hyers–Ulam stability and uniform exponential stability of 2-periodic linear nonautonomous systems. Adv. Differ. Equ. 2017, 1–7 (2017)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Zada, A., Faisal, S., Li, Y.: On the Hyers-Ulam stability of first order impulsive delay differential equations. J. Funct. Spaces 2016, 8164978 (2016).  https://doi.org/10.1155/2016/8164978 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Zada, A., Shah, O., Shah, R.: Hyers–Ulam stability of non-autonomous systems in terms of boundedness of Cauchy problems. Appl. Math. Comput. 271, 512–518 (2015)MathSciNetzbMATHGoogle Scholar
  44. 44.
    Zavalishchin, S., Sesekin, A.: Impulsive Processes: Models and Applications. Nauka, Moscow (1991)zbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsCOMSATS University Islamabad, Sahiwal CampusSahiwalPakistan
  2. 2.Department of Mathematics and StatisticsUniversity of SwatSwatPakistan
  3. 3.Department of MathematicsUniversity of MalakandDir(L)Pakistan

Personalised recommendations