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On the Bielecki–Ulam’s Type Stability Results of First Order Non-linear Impulsive Delay Dynamic Systems on Time Scales

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Abstract

In this manuscript, we present the existence and uniqueness of solution of first order non-linear impulsive delay dynamic systems on time scales, with the help of fixed point approach. Further, we investigate the Bielecki–Ulam–Hyers stability and Bielecki–Ulam–Hyers–Rassias stability of the proposed model. We use Grönwall’s inequality on time scales, Banach contraction principle, Picard operator and Lipschitz condition as a basic tools to develop our main results. Also, two examples are provided to illustrate the obtained results.

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References

  1. Agarwal, R.P., Awan, A.S., ÓRegan, D., Younus, A.: Linear impulsive Volterra integro–dynamic system on time scales. Adv. Differ. Equ., 2014(6)(2014), 1–17

  2. Ahmad, M., Jiang, J., Zada, A., Shah, S.O., Xu, J.: Analysis of coupled system of implicit fractional differential equations involving Katugampola–Caputo fractional derivative. Complexity 2020, 1–11 (2020)

    MATH  Google Scholar 

  3. Ahmad, M., Zada, A., Alzabut, J.: Stability analysis of a nonlinear coupled implicit switched singular fractional differential equations with \(p\)-Laplacian. Adv. Differ. Equ. 2019(436), 1–22 (2019)

    Google Scholar 

  4. Ahmad, M., Zada, A., Alzabut, J.: Hyers–Ulam stability of a coupled system of fractional differential equations of Hilfer–Hadamard type. Demonstr. Math. 52, 283–295 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  5. Alsina, C., Ger, R.: On some inequalities and stability results related to the exponential function. J. Inequal. Appl. 2, 373–380 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Alzabut, J., Mohammadaliee, B., Samei, M.E.: Solutions of two fractional q-integro-differential equations under sum and integral boundary value conditions on a time scale. Adv. Differ. Equ. 2020(304), 1–33 (2020)

    MathSciNet  Google Scholar 

  7. András, S., Mészáros, A.R.: Ulam–Hyers stability of dynamic equations on time scales via Picard operators. Appl. Math. Comput. 219(9), 4853–4864 (2013)

    MathSciNet  MATH  Google Scholar 

  8. Bainov, D.D., Dishliev, A.: Population dynamics control in regard to minimizing the time necessary for the regeneration of a biomass taken away from the population. Comp. Rend. Bulg. Scie. 42, 29–32 (1989)

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  10. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)

    Book  MATH  Google Scholar 

  11. Bohner, M., Peterson, A.: Advances in Dynamics Equations on Time Scales. Birkhäuser, Boston (2003)

    Book  MATH  Google Scholar 

  12. Dachunha, J.J.: Stability for time varying linear dynamic systems on time scales. J. Comput. Appl. Math. 176(2), 381–410 (2005)

    Article  MathSciNet  Google Scholar 

  13. Hamza, A., Oraby, K.M.: Stability of abstract dynamic equations on time scales. Adv. Differ. Equ. 2012(143), 1–15 (2012)

    MathSciNet  MATH  Google Scholar 

  14. Hilger, S.: Analysis on measure chains: a unified approach to continuous and discrete calculus. Result Math. 18, 18–56 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27(4), 222–224 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  16. Jung, S.-M.: Hyers–Ulam stability of linear differential equations of first order. Appl. Math. Lett. 17(10), 1135–1140 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, NewYork, 48 (2011)

  18. Li, Y., Shen, Y.: Hyers–Ulam stability of linear differential equations of second order. Appl. Math. Lett. 23(3), 306–309 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(86), 1–18 (2019)

    MathSciNet  MATH  Google Scholar 

  20. Lupulescu, V., Zada, A.: Linear impulsive dynamic systems on time scales. Electron. J. Qual. Theory Differ. Equ. (11)(2010), 1–30

  21. Matar, M.M., Skhail, E.S.A., Alzabut, J.: On solvability of nonlinear fractional differential systems involving nonlocal initial conditions. Math. Meth. Appl. Sci., 2019, 1–12. https://doi.org/10.1002/mma.5910

  22. Nenov, S.I.: Impulsive controllability and optimization problems in population dynamics. Nonlinear Anal. Theory Methods Appl. 36(7), 881–890 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  23. Obłoza, M.: Hyers stability of the linear differential equation. Rocznik Nauk.-Dydakt. Prace Mat., (13)(1993), 259–270

  24. Obłoza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Nauk.-Dydakt. Prace Mat. 14, 141–146 (1997)

    MathSciNet  MATH  Google Scholar 

  25. Pötzsche, C., Siegmund, S., Wirth, F.: A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete Contin. Dyn. Syst. 9, 1223–1241 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rassias, T.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  27. Rizwan, R., Zada, A.: Nonlinear impulsive Langevin equation with mixed derivatives. Math. Meth. Appl. Sci. 43(1), 427–442 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  28. Rizwan, R., Zada, A., Wang, X.: Stability analysis of non linear implicit fractional Langevin equation with non-instantaneous impulses. Adv. Differ. Equ. 2019(85), 1–31 (2019)

    MATH  Google Scholar 

  29. Seemab, A., Rehman, M.U., Alzabut, J., Hamdi, A.: On the existence of positive solutions for generalized fractional boundary value problems. Bound. Value. Probl. 2019(186), 1–20 (2019)

    MathSciNet  Google Scholar 

  30. Shah, S.O., Zada, A., Hamza, A.E.: Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales. Qual. Theory Dyn. Syst. 18(3), 825–840 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shah, S.O., Zada, A.: On the stability analysis of non-linear Hammerstein impulsive integro-dynamic system on time scales with delay. Punjab Univ. J. Math. 51(7), 89–98 (2019)

    MathSciNet  Google Scholar 

  32. Shah, S.O., Zada, A.: Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales. Appl. Math. Comput. 359, 202–213 (2019)

    MathSciNet  MATH  Google Scholar 

  33. Ulam, S.M.: A Collection of the Mathematical Problems. Interscience Publisheres, New York (1960)

    MATH  Google Scholar 

  34. Ulam, S.M.: Problem in Modern Mathematics, Science Editions. Wiley, New York (1964)

    Google Scholar 

  35. Wang, J.R., Fečkan, M., Tian, Y.: Stability analysis for a general class of non-instantaneous impulsive differential equations. Mediterr. J. Math. 14(46), 1–21 (2017)

    MathSciNet  MATH  Google Scholar 

  36. Wang, J.R., Fečkan, M., Zhou, Y.: Ulam’s type stability of impulsive ordinary differential equations. J. Math. Anal. Appl. 395(1), 258–264 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wang, J.R., Fečkan, M., Zhou, Y.: On the stability of first order impulsive evolution equations. Opuscula Math. 34(3), 639–657 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wang, J.R., Li, X.: A uniform method to Ulam–Hyers stability for some linear fractional equations. Mediterr. J. Math. 13, 625–635 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wang, J.R., Zhang, Y.: A class of nonlinear differential equations with fractional integrable impulses. Commun. Nonlinear Sci. Numer. Simul. 19(9), 3001–3010 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Younus, A., O’Regan, D., Yasmin, N., Mirza, S.: Stability criteria for nonlinear Volterra integro-dynamic systems. Appl. Math. Inf. Sci. 11(5), 1509–1517 (2017)

    Article  MathSciNet  Google Scholar 

  41. Zada, A., Alzabut, J., Waheed, H., Popa, I.L.: Ulam–Hyers stability of impulsive integrodifferential equations with Riemann–Liouville boundary conditions. Adv. Differ. Equ. 2020(64), 1–50 (2020)

    MathSciNet  Google Scholar 

  42. Zada, A., Pervaiz, B., Shah, S.O., Xu, J.: Stability analysis of first-order impulsive nonautonomous system on timescales. Math. Meth. Appl. Sci. 43(8), 5097–5113 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  44. Zada, A., Shah, S.O.: Hyers–Ulam stability of first-order non-linear delay differential equations with fractional integrable impulses. Haceteppe J. Math. Stat. 47(5), 1196–1205 (2018)

    MathSciNet  Google Scholar 

  45. Zada, A., Waheed, H., Alzabut, J., Wang, X.: Existence and stability of impulsive coupled system of fractional integrodifferential equations. Demonstr. Math. 52(1), 296–335 (2019)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors express their sincere gratitude to the Editor and referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results.

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Authors and Affiliations

  1. Department of Physical and Numerical Sciences, Qurtuba University of Science and Information Technology Peshawar, Dera Ismail Khan, Pakistan

    Syed Omar Shah, Muzamil Muzamil & Muhammad Tayyab

  2. Department of Mathematics, University of Peshawar, Peshawar, 25000, Pakistan

    Akbar Zada

  3. Department of Mathematics, University of Buner, Buner, Pakistan

    Rizwan Rizwan

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  1. Syed Omar Shah
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  2. Akbar Zada
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  3. Muzamil Muzamil
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Correspondence to Syed Omar Shah.

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Shah, S.O., Zada, A., Muzamil, M. et al. On the Bielecki–Ulam’s Type Stability Results of First Order Non-linear Impulsive Delay Dynamic Systems on Time Scales. Qual. Theory Dyn. Syst. 19, 98 (2020). https://doi.org/10.1007/s12346-020-00436-8

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