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Bielecki–Ulam’s Types Stability Analysis of Hammerstein and Mixed Integro–Dynamic Systems of Non–Linear Form with Instantaneous Impulses on Time Scales

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Abstract

In this paper, the stability in terms of Bielecki–Ulam–Hyers and stability in terms of Bielecki–Ulam–Hyers–Rassias of non–linear impulsive Hammerstein integro–dynamic system with delay and non–linear impulsive mixed integro–dynamic system on time scales are achieved by utilizing fixed point approach along with Lipschitz condition and Gronwall’s inequality. Examples are also provided for the verification of the 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), 1–17 (2014)

    MathSciNet  Google Scholar 

  2. 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 

  3. Alzabut, J.O., 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  MATH  Google Scholar 

  4. 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 

  5. 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 

  6. Bainov, D.D., Simenov, P.S.: Systems with impulse effect stability theory and applications. Ellis Horwood Limited, Chichester, UK (1989)

    Google Scholar 

  7. Bohner, M., Li, T.: Kamenev-type criteria for nonlinear damped dynamic equations. Sci. China Math. 58(7), 1445–1452 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Bohner, M., Peterson, A.: Dynamic equations on time scales: an introduction with applications. Birkhäuser, Boston, Mass, USA (2001)

    MATH  Google Scholar 

  9. Bohner, M., Peterson, A.: Advances in dynamics equations on time scales. Birkhäuser, Boston, Mass, USA (2003)

    MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  11. Georgieva, A., Kostadinov, S., Stamov, G.T., Alzabut, J.O.: On \(L_p(k)\)-equivalence of impulsive differential equations and its applications to partial impulsive differential equations. Adv. Differ. Equ. 2012(144), 1–12 (2012)

    MATH  Google Scholar 

  12. 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 

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

    MathSciNet  MATH  Google Scholar 

  14. Hyers, D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A. 27(4), 222–224 (1941)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Jung, S.-M.: Hyers–Ulam–Rassias stability of functional equations in nonlinear analysis, Springer Optim. Appl., Springer, NewYork, 48, (2011)

  17. Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70(3), 1–18 (2019)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  19. Li, T., Viglialoro, G.: Boundedness for a nonlocal reaction chemotaxis model even in the attraction-dominated regime. Differ. Integral Equ. 34(5–6), 315–336 (2021)

    MathSciNet  MATH  Google Scholar 

  20. Li, T., Zada, A.: Connections between Hyers-Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equ. 2016(153), 1–8 (2016)

    MathSciNet  MATH  Google Scholar 

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

  22. Marras, M., Viglialoro, G.: Blow-up time of a general Keller-Segel system with source and damping terms. Comptes Rendus de L’Academie Bulgare des Sci. 69(6), 687–696 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

  25. 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 

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

    MathSciNet  MATH  Google Scholar 

  27. Pratap, A., Raja, R., Alzabut, J.O., Cao, J., Rajchakit, G., Huang, C.: Mittag-Leffler stability and adaptive impulsive synchronization of fractional order neural networks in quaternion field. Math. Methods Appl. Sci. 43(10), 6223–6253 (2020)

    MathSciNet  MATH  Google Scholar 

  28. Qiu, Y.-C., Zada, A., Tang, S., Li, T.: Existence of nonoscillatory solutions to nonlinear third-order neutral dynamic equations on time scales. J. Nonlinear Sci. Appl. 10(8), 4352–4363 (2017)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  30. Saker, S.H., Alzabut, J.O.: On the impulsive delay Hematopoiesis model with periodic coefficients. Rocky. Mt. J. Math. 39(5), 1657–1688 (2009)

    MathSciNet  MATH  Google Scholar 

  31. Senthilraj, S., Saravanakumar, T., Raja, R., Alzabut, J.O.: Delay-dependent passivity analysis of nondeterministic genetic regulatory networks with leakage and distributed delays against impulsive perturbations. Adv. Differ. Equ. 2021(353), 1–26 (2021)

    MathSciNet  MATH  Google Scholar 

  32. 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 

  33. 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 

  34. 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)

    MathSciNet  MATH  Google Scholar 

  35. Shah, S.O., Zada, A., Muzamil, M., Tayyab, M., Rizwan, R.: 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), 1–18 (2020)

    MathSciNet  MATH  Google Scholar 

  36. Stamov, G Tr., Alzabut, J.O., Atanasov, P., Stamov, A.G.: Almost periodic solutions for an impulsive delay model of price fluctuations in commodity markets. Nonlinear Anal. Real World Appl. 12(6), 3170–3176 (2011)

    MathSciNet  MATH  Google Scholar 

  37. Stamov, G.T., Stamova, I.M., Alzabut, J.O.: Existence of almost periodic solutions for strongly stable nonlinear impulsive differential-difference equations. Nonlinear Anal. Hybrid Syst. 6(2), 818–823 (2012)

    MathSciNet  MATH  Google Scholar 

  38. Ulam, S.M.: A collection of the mathematical problems. Interscience Publisheres, New York- London (1960)

    MATH  Google Scholar 

  39. Ulam, S.M.: Problem in modern mathematics, Science Editions. J. Wiley and Sons Inc, New York (1964)

    Google Scholar 

  40. Vinodkumar, A., Senthilkumar, T., Hariharan, S., Alzabut, J.O.: Exponential stabilization of fixed and random time impulsive delay differential system with applications. Math. Biosci. Eng. 18(3), 2384–2400 (2021)

    MathSciNet  MATH  Google Scholar 

  41. 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 

  42. 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)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  45. Wang, X., Rizwan, R., Lee, J.R., Zada, A., Shah, S.O.: Existence, uniqueness and Ulam’s stabilities for a class of implicit impulsive Langevin equation with Hilfer fractional derivatives. AIMS Math. 6(5), 4915–4929 (2021)

    MathSciNet  MATH  Google Scholar 

  46. 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)

    MathSciNet  MATH  Google Scholar 

  47. 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)

    MathSciNet  Google Scholar 

  48. Zada, A., Ali, S., Li, T.: Analysis of a new class of impulsive implicit sequential fractional differential equations. Int. J. Nonlinear Sci. Numer. Simul. 21(6), 571–587 (2020)

    MathSciNet  MATH  Google Scholar 

  49. 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 

  50. Zada, A., Pervaiz, B., Subramanian, M., Popa, I.: Finite time stability for nonsingular impulsive first order delay differential systems. Appl. Math. Comput. 421, 126943 (2022)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

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

    Syed Omar Shah, Qayyum Ullah Khan, Iftikhar Ullah & Ibrar Ullah

  2. Department of Mathematics, Faculty of Sciences, Van Yuzuncu Yil University, 65080, Van, Turkey

    Cemil Tunç

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

    Rizwan Rizwan

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

    Akbar Zada

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  1. Syed Omar Shah
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Shah, S.O., Tunç, C., Rizwan, R. et al. Bielecki–Ulam’s Types Stability Analysis of Hammerstein and Mixed Integro–Dynamic Systems of Non–Linear Form with Instantaneous Impulses on Time Scales. Qual. Theory Dyn. Syst. 21, 107 (2022). https://doi.org/10.1007/s12346-022-00639-1

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