Skip to main content
SpringerLink
Log in

On mild solutions of fractional impulsive differential systems of Sobolev type with fractional nonlocal conditions

  • Original Research
  • Published:
Mathematical Sciences Aims and scope Submit manuscript

Abstract

This paper concerns the application of the monotone iterative technique in conjunction with the lower and upper solution techniques to investigate the existence of mild solutions and their uniqueness for fractional impulsive differential systems of the Sobolev type with fractional order nonlocal conditions. To obtain the adequate requirements, noncompactness estimates and the generalized Gronwall inequality are utilized.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Amar, D., Delfirm, F.M.T.: Sobolev type fractional abstract evolution equations with nonlocal conditions and optimalmulti-controls. Appl. Math. Comput. 245, 74–85 (2014)

    MathSciNet  Google Scholar 

  2. Balachandran, K., Kiruthika, S., Trujillo, J.J.: On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces. Comput. Math. Appl. 62, 1157–1165 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Banas, J., Goebel, K.: Measure of Noncompactness in Banach Space. Marcal Dekker Inc., New York (1980)

    MATH  Google Scholar 

  4. Barenblat, G., Zheltor, J., Kochiva, I.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960)

    Google Scholar 

  5. Benchohra, M., Henderson, J., Ntouyas, S.K.: Impulsive Differential Equations and Inclusions. Contemporary Mathematics and its Applications, vol. 2. Hindawi Publishing Corporation, New York (2006)

    MATH  Google Scholar 

  6. Bhaskar, T.G., Lakshmikantham, V., Devi, J.V.: Monotone iterative technique for functional differential equations with retardation and anticipation. Nonlinear Anal. 66(10), 2237–2242 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Bothe, D.: Multivalued perturbations of m-accretive differential inclusions. Isr. J. Math. 108, 109–138 (1998)

    MathSciNet  MATH  Google Scholar 

  8. Brill, H.: A semilinear Sobolev evolution equation in Banach space. J. Differ. Equ. 24, 412–425 (1977)

    MathSciNet  MATH  Google Scholar 

  9. Byszewski, L.: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl. 162, 494–505 (1991)

    MathSciNet  MATH  Google Scholar 

  10. Byszewski, L., Akca, H.: Existence of solutions of a nonlinear functional differential evolution nonlocal problem. Nonlinear Anal. 34, 65–72 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Byszewski, L., Lakshmikantham, V.: Theorem about the existence and uniqueness of a solution of a nonlocal abstract Cauchy problem in a Banach space. Appl. Anal. 40(1), 11–19 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Chen, P.J., Curtin, M.E.: On a theory of heat conduction involving two temperatures. Z. Angew. Math. Phys. 19, 614–627 (1968)

    Google Scholar 

  13. Chen, P., Li, Y.: Mixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces. Nonlinear Anal. 74, 3578–3588 (2011)

    MathSciNet  MATH  Google Scholar 

  14. Chengxuan, X.I.E., Xiaoxiao, X.I.A., Yones, E.A., Behnaz, F., Hossein J., Shuchun, W.: The Numerical Strategy of Tempered Fractional Derivative in European Double Barrier Option, Fractals, vol. 16, p. 22. September 7, 2021, 0218–348X

  15. Chen, P., Mu, J.: Monotone iterative method for semilinear impulsive evolution equations of mixed type in Banach spaces. Electron. J. Differ. Equ. 14, 1–13 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Chaudhary, R.: Monotone iterative technique for Sobolev type fractional integro-differential equations with fractional nonlocal conditions. Rendiconti del Circolo Matematico di Palermo Series 2(69), 925–937 (2020)

    MathSciNet  MATH  Google Scholar 

  17. Chaudhary, R., Pandey, D.N.: Monotone iterative technique for neutral fractional differential equation with infinite delay. Math. Methods Appl. Sci. 39(15), 4642–4653 (2016)

    MathSciNet  MATH  Google Scholar 

  18. Chaudhary, R., Pandey, D.N.: Monotone iterative technique for impulsive Riemann–Liouville fractional differential equations. Filomat 39(9), 3381–3395 (2018)

    MathSciNet  MATH  Google Scholar 

  19. Dabas, J., Chauhan, A.: Existence and uniqueness of mild solution for an impulsive neutral fractional integro-differential equation with infinite delay. Math. Comput. Model. 57(3–4), 754–763 (2013)

    MathSciNet  MATH  Google Scholar 

  20. Debbouche, A., Torres, D.F.M.: Sobolev type fractional dynamic equations and optimal multi-integral controls with fractional nonlocal conditions. Fract. Calc. Appl. Anal. 18(1), 95–121 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)

    MATH  Google Scholar 

  22. Deng, K.: Exponential decay of solutions of semilinear parabolic equations with nonlocal initial conditions. J. Math. Anal. Appl. 179, 630–637 (1993)

    MathSciNet  MATH  Google Scholar 

  23. Esmaeelzade Aghdam, Y., Mesgarani, H., Moremedi, G.M., Khoshkhahtinat, M.: High-accuracy numerical scheme for solving the space-time fractional advection–diffusion equation with convergence analysis. Alex. Eng. J. 61(1), 217–225 (2021)

    Google Scholar 

  24. Feckan, M., Zhou, Y., Wang, J.: On the concept and existence of solution for impulsive fractional differentail equations. Commun. Nonlinear Sci. Number Simul. 17(7), 3050–3060 (2012)

    MATH  Google Scholar 

  25. Feckan, M., Zhou, Y., Wang, J.: Response to “Comments on the concept of existence of solution for impulsive fractional differential equations [Commun Nonlinear Sci Numer Simul 2014; 19:401-3.]’’. Commun. Nonlinear Sci. Numer. Simul. 19(12), 4213–4215 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Haiping, Y., Jianming, G., Yongsheng, D.: A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 328(2), 1075–1081 (2007)

    MathSciNet  MATH  Google Scholar 

  27. Heinz, H.: On the behavior of measures of noncompactness with respect to differentiation and integration of vector valued functions. Nonlinear Anal. 7(12), 1351–1371 (1983)

    MathSciNet  MATH  Google Scholar 

  28. Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45, 765–771 (2006)

    Google Scholar 

  29. Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    MATH  Google Scholar 

  30. Hristova, S.G., Bainov, D.D.: Applications of monotone iterative techniques of V. Lakshmikantham to the solution of the initial value problem for functional differential equations. Le Math. 44, 227–236 (1989)

    MathSciNet  Google Scholar 

  31. Kamaljeet, B.D.: Monotone iterative technique for nonlocal fractional differential equations with finite delay in a Banach space. Electron. J. Qual. Theory Differ. Equ. 3, 1–16 (2015)

    MathSciNet  Google Scholar 

  32. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, vol. 204. North-Holland Mathematics Studies (2006)

  33. Lakshmikantham, V., Vatsala, A.S.: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 21(8), 828–834 (2008)

    MathSciNet  MATH  Google Scholar 

  34. Lakshmikantham, V., Bainov, D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989)

    Google Scholar 

  35. Li, Y., Liu, Z.: Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces. Nonlinear Anal. 66(1), 83–92 (2007)

    MathSciNet  MATH  Google Scholar 

  36. Li, F., Liang, J., Xu, H.-K.: Existence of mild solutions for fractional integro differential equations of Sobolev type with nonlocal conditions. J. Math. Anal. Appl. 391, 510–525 (2012)

    MATH  Google Scholar 

  37. Magin, R.L.: Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl. 59, 1586–1593 (2010)

    MathSciNet  MATH  Google Scholar 

  38. Meral, F.C., Royston, T.J., Magin, R.: Fractional calculus in viscoelasticity: an experimental study. Commun. Nonlinear Sci. Numer. Simul. 15, 939–945 (2010)

    MathSciNet  MATH  Google Scholar 

  39. Mesgarani, H., Esmaeelzade Aghdam, Y., Tavakoli, H.: Numerical simulation to solve two-dimensional temporal-space fractional Bloch–Torrey equation taken of the spin magnetic moment diffusion. Int. J. Appl. Comput. Math. 7, 94 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Mesgaran, H., Beiranvand, A., Esmaeelzade Aghdam, Y.: The impact of the Chebyshev collocation method on solutions of the time-fractional Black-Scholes. Math. Sci. 15, 137–143 (2021)

    MathSciNet  MATH  Google Scholar 

  41. Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)

    MATH  Google Scholar 

  42. Sun, Y.-F., Zeng, Z., Song, J.: Quasilinear iterative method for the boundary value problem of nonlinear fractional differential equation. Numer. Algebra Control Optim. 10(2), 157–164 (2020)

    MathSciNet  MATH  Google Scholar 

  43. Wang, P.G., Tian, S.-H., Wu, Y.-H.: Monotone iterative method for first-order functional difference equations with nonlinear boundary value conditions. Appl. Math. Comput. 203, 266–272 (2008)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to express our gratitude to the anonymous referee for his insightful comments and ideas for improving the manuscript.

Author information

Authors and Affiliations

  1. Department of Mathematics, KPR Institute of Engineering and Technology, Coimbatore, Tamil Nadu, 641407, India

    K. Karthikeyan

  2. Department of Mathematics, Nandha Engineering College, Erode, Tamil Nadu, 52, India

    G. S. Murugapandian

  3. Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot, Binh Duong, Vietnam

    Z. Hammouch

  4. Department of Medical Research, China Medical University Hospital, Taichung, Taiwan

    Z. Hammouch

  5. Department of Sciences, Ecole Normale Superieure of Meknes, Moulay Ismail University, Mekens, Morocco

    Z. Hammouch

Authors
  1. K. Karthikeyan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. S. Murugapandian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Z. Hammouch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Z. Hammouch.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Karthikeyan, K., Murugapandian, G.S. & Hammouch, Z. On mild solutions of fractional impulsive differential systems of Sobolev type with fractional nonlocal conditions. Math Sci 17, 285–295 (2023). https://doi.org/10.1007/s40096-022-00469-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40096-022-00469-x

Keywords

Mathematics Subject Classification

Search

Navigation