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Hyers–Ulam Stability for a Coupled System of Fractional Differential Equation With p-Laplacian Operator Having Integral Boundary Conditions

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Abstract

In this article we explore the existence, uniqueness, and stability for a coupled symmetric system of fractional differential equation with nonlinear p-Laplacian operator. Existence and uniqueness results are obtained by using the matrix eigenvalue method. Further, we study different types of Hyers–Ulam stability. In the last section an example concerning the proposed problem is presented.

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References

  1. Alam, M., Zada, A.: Implementation of \(q\)-calculus on \(q\)-integro-differential equation involving anti-periodic boundary conditions with three criteria. Chaos Solitons Fractals 154, 111625 (2020)

    Article  MathSciNet  Google Scholar 

  2. Bagley, R.L., Torvik, P.J.: On the fractional calculus model of viscoelastic behavior. J. Rheol. 30(1), 133–155 (1986)

    Article  Google Scholar 

  3. Bai, C.: Existence and uniqueness of solutions for fractional boundary value problems with \(p\)-Laplacian operator. Adv. Differ. Equ. 2018(1), 1–12 (2018)

    Article  MathSciNet  Google Scholar 

  4. Bai, Z., Lv, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation. J. Math. Anal. Appl. 311, 495–505 (2005)

    Article  MathSciNet  Google Scholar 

  5. Balachandran, K., Govindaraj, V.: Numerical controllability of fractional dynamical systems. Optimization 63(8), 1267–1279 (2014)

    Article  MathSciNet  Google Scholar 

  6. Diethelm, K., Freed, A.D.: On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity. In: Keil, F., Mackens, W., Vob, H., Werther, J. (eds.) Scientific Computing in Chemical Engineering II, pp. 217–224. Springer, Berlin (1999)

    Chapter  Google Scholar 

  7. Gao, L., Wang, D., Wang, G.: Further results on exponential stability for impulsive switched nonlinear time delay systems with delayed impulse effects. Appl. Math. Comput. 268, 186–200 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Glöckle, W.G., Nonnenmacher, T.F.: A fractional calculus approach to self-similar protein dynamics. Biophys. J. 68(1), 46–53 (1995)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Isaia, F.: On a nonlinear integral equation without compactness. Acta Math. Univ. Comenian. 75, 233–240 (2006)

    MathSciNet  MATH  Google Scholar 

  11. Khamessi, B., Hamiaz, A.: Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with \(p\)-Laplacian operator. J. Taibah Univ. Sci. 13(1), 370–376 (2019)

    Article  Google Scholar 

  12. Khan, A., Li, Y. Shah, K. Khan, T.S.: On coupled \(p\)-Laplacian fractional differential equations with nonlinear boundary conditions. Complexity, 2017, (2017)

  13. Khan, H., Chen, W., Khan, A., Khan, T.S., Al-Madlal, Q.M.: Hyers-Ulam stability and existence criteria for coupled fractional differential equations involving \(p\)-Laplacian operator. Adv. Differ. Equ. 455, 1–16 (2018)

    MathSciNet  MATH  Google Scholar 

  14. Khan, A., Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T.: Existence and Hyers–Ulam stability for a nonlinear singular fractional differential equations with Mittag–Leffler kernel. Chaos Solitons Fractals 127, 422–427 (2019)

    Article  MathSciNet  Google Scholar 

  15. Khan, A., Gómez-Aguilar, J.F., Abdeljawad, T., Khan, H.: Stability and numerical simulation of a fractional order plant-nectar-pollinator model. Alex. Eng. J. 59, 49–59 (2020)

    Article  Google Scholar 

  16. Khan, H., Tunc, C., Khan, A.: Green function’s properties and existence theorems for nonlinear singular-delay-fractional differential equations. Am. Inst. Math. Sci. 13, 2475–2487 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Khan, H., Gómez-Aguilar, J.F., Abdeljawad, T., Khan, A.: Existence results and stability criteria for ABC-fuzzy-volterra integro-differential equation. Fractals 28, 2040048 (2020)

    Article  Google Scholar 

  18. Kilbas, A.A., Srivastava, H.M. Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies; Elsevier Science B.V.: Amsterdam, The Netherlands, Volume 204 (2006)

  19. Leibenson, L.S.: General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk Kirg. SSR 9(1), 7–10 (1983)

    MathSciNet  Google Scholar 

  20. Li, F., Bao, Y.: Uniform stability of the solution for a Memory-type elasticity system with nonhomogeneous boundary control condition. J. Dyn. Control Syst. 23(2), 301–315 (2017)

    Article  MathSciNet  Google Scholar 

  21. Liang, Z., Han, X., Li, A.: Some properties and applications related to the \((2, p)\)-Laplacian operator. Bound. Val. Probl. 2016(1), 1–17 (2016)

    Article  MathSciNet  Google Scholar 

  22. Lu, H., Han, Z., Sun, S.: Multiplicity of positive solutions for Sturm-Liouville boundary value problems of fractional differential equations with \(p\)-Laplacian. Bound. Value Probl. 2014(1), 1–17 (2014)

    Article  MathSciNet  Google Scholar 

  23. Magin, R.L.: Fractional Calculus in Bioengineering. Begell House Redding, Oxford (2006)

    Google Scholar 

  24. Magin, R., Ortigueira, M.D., Podlubny, I., Trujillo, J.J.: On the fractional signals and systems. Signal Proc. 91(3), 350–371 (2011)

    Article  Google Scholar 

  25. Matar, M.M., Lubbad, A.A., Alzabut, J.: On \(p\)-Laplacian boundary value problems involving Caputo-Katugampula fractional derivatives. Math. Methods Appl. Sci. 2020, 1–18 (2020)

    Google Scholar 

  26. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)

    MATH  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, New York (1999)

    MATH  Google Scholar 

  28. Prasad, K.R., Krushna, B.M.B.: Multiple positive solutions for a coupled system of \(p\)-Laplacian fractional order two-point boundary value problems. Int. J. Differ. Equ., 2014 (2014)

  29. Shao, J., Guo, B.: Existence of solutions and Hyers–Ulam stability for a coupled system of nonlinear fractional differential equations with \(p\)-Laplacian operator. Symmetry 13(7), 1160 (2021)

    Article  Google Scholar 

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

    MATH  Google Scholar 

  31. Urs, C.: Ulam–Hyers stability for coupled fixed points of contractive type operators. J. Nonlinear Sci. Appl. 6(2), 124–136 (2013)

    Article  MathSciNet  Google Scholar 

  32. Urs, C.: Coupled fixed point theorem and applications to periodic boundary value problem. Miskolic Math Notes 14, 323–333 (2013)

    Article  MathSciNet  Google Scholar 

  33. Waheed, H., Zada, A., Xu, J.: Well-posedness and Hyers–Ulam results for a class of impulsive fractional evolution equations. Math. Methods Appl. Sci. 44(1), 749–771 (2021)

    Article  MathSciNet  Google Scholar 

  34. Wang, J.R., Zada, A., Waheed, H.: Stability analysis of a coupled system of nonlinear implicit fractional anti-periodic boundary value problem. Math. Methods Appl. Sci. 42(18), 6706–6732 (2019)

    Article  MathSciNet  Google Scholar 

  35. Wenchang, T., Wenxiao, P., Mingyu, X.: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates. Int. J. Nonlinear Mech. 38(5), 645–650 (2003)

    Article  Google Scholar 

  36. Yang, Z.Y., Blanke, M.: A unified approach to controllability analysis for hybrid systems. Nonlinear Anal. Hybrid Syst. 1(2), 212–222 (2007)

    Article  MathSciNet  Google Scholar 

  37. Zada, A., Waheed, H.: Stability analysis of implicit fractional differential equation with anti-periodic integral boundary value problem. Ann. Univ. Paedagog. Crac. Stud. Math. 19(1), 5–25 (2019)

    MathSciNet  Google Scholar 

  38. 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  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  40. Zada, A., Pervaiz, B., Subramanian, M., Popa, I.L.: 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 Mathematics, University of Peshawar, Peshawar, 25000, Pakistan

    Hira Waheed & Akbar Zada

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

    Rizwan Rizwan

  3. Department of Exact Science and Engineering, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, 510009, Romania

    Ioan-Lucian Popa

Authors
  1. Hira Waheed
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  2. Akbar Zada
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  3. Rizwan Rizwan
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  4. Ioan-Lucian Popa
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All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript.

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Correspondence to Hira Waheed.

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Waheed, H., Zada, A., Rizwan, R. et al. Hyers–Ulam Stability for a Coupled System of Fractional Differential Equation With p-Laplacian Operator Having Integral Boundary Conditions. Qual. Theory Dyn. Syst. 21, 92 (2022). https://doi.org/10.1007/s12346-022-00624-8

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  • DOI: https://doi.org/10.1007/s12346-022-00624-8

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