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Existence and Hyers–Ulam Stability of Jerk-Type Caputo and Hadamard Mixed Fractional Differential Equations

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Abstract

This article is concerned with existence of mild solutions for jerk-type fractional differential equations in the sense of Hadamard and Caputo fractional derivatives with separated boundary conditions. For the uniqueness of mild solutions in both cases, Banach contraction principle are followed. Moreover, at least one mild solution of jerk-type Caputo–Hadamard and Hadamard–Caputo fractional differential equations can be analyzed using Krasnoselskii’s and Leray–Schauder fixed point theorems. Hyers–Ulam stability and its generalized case for both type of mentioned jerk-type problems can be find out with the help of some conditions and definitions. For the illustration of main results, an example is provided.

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References

  1. Podlubny, I.: Fractional Differential Equations. Elsevier, Amsterdam (1999)

    Google Scholar 

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

  3. Ahmad, B., Alsaedi, A., Ntouyas, S.K., Tariboon, J.: Hadamard-Type Fractional Differential Equations. Inclusions and Inequalities. Springer, Berlin (2017)

    Google Scholar 

  4. Diethelm, K.: The Analysis of Fractional Differential Equations: An Application—Oriented Exposition Using Differential Operators of Caputo Type. Springer Science and Business Media (2010)

  5. Maraaba, T., Baleanu, D., Jarad, F.: Existence and uniqueness theorem for a class of delay differential equations with left and right Caputo fractional derivatives. J. Math. Phys. 49(8), 1–11 (2008)

    MathSciNet  Google Scholar 

  6. Wei, Z., Li, Q., Che, J.: Initial value problems for fractional differential equations involving Riemann–Liouville sequential fractional derivative. J. Math. Ana. Appl. 1, 260–272 (2010)

    MathSciNet  Google Scholar 

  7. Pooseh, S., Almeida, R., Torres, D.F.: Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative. Numer. Func. Anal. Optim. 3, 301–319 (2012)

    MathSciNet  Google Scholar 

  8. Zada, A., Alam, M., Riaz, U.: Analysis of q-fractional implicit boundary value problems having Stieltjes integral conditions. Math. Meth. Appl. Sci. 44(6), 4381–4413 (2020)

    MathSciNet  Google Scholar 

  9. Alam, M., Zada, A., Riaz, U.: On a coupled impulsive fractional integrodifferential system with Hadamard derivatives. Qual. Theory Dyn. Syst. 21(8), 1–31 (2021)

    MathSciNet  Google Scholar 

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

  11. Kilbas, A. A., Srivastava, H. M., Trujillo, J. J.: Theory and Applications of the Fractional Differential Equations. North–Holland Mathematics Studies, Vol. 204 (2006)

  12. Zhang, X., Liu, L., Wu, Y.: Multiple positive solutions of a singular fractional differential equation with negatively perturbed term. Math. Comput. Model. 55, 1263–1274 (2012)

    MathSciNet  Google Scholar 

  13. Zhang, X., Wu, Y., Caccetta, L.: Nonlocal fractional order differential equations with changing-sign singular perturbation. Appl. Math. Model. 39, 6543–6552 (2015)

    MathSciNet  Google Scholar 

  14. Zhou, Y., Wang, J., Zhang, L.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2016)

    Google Scholar 

  15. Bai, Z., Sun, W.: Existence and multiplicity of positive solutions for singular fractional boundary value problems. Comput. Math. Appl. 63(9), 1369–1281 (2012)

    MathSciNet  Google Scholar 

  16. Tariboon, J., Cuntavepanit, A., Ntouyas, S.K., Nithiarayaphaks, W.: Separated boundary value problems of sequential Caputo and Hadamard fractional differential equations. J. Funct. Spaces 2018, 1–8 (2018)

    MathSciNet  Google Scholar 

  17. Ahmad, B., Ntouyas, S.K.: Fractional differential inclusions with fractional separated boundary conditions. Fract. Calc. Appl. Anal. 15(3), 362–382 (2012)

    MathSciNet  Google Scholar 

  18. Ahmad, B., Ntouyas, S.K., Alsaed, A., Shammakh, W., Agarwal, R.P.: Existence theory for fractional differential equations with non-separated type nonlocal multi-point and multi-strip boundary conditions. Adv. Differ. Equ. 2018(89), 1–20 (2018)

    MathSciNet  Google Scholar 

  19. Agarwal, R.P., Ahmad, B., Alsaedi, A.: Fractional-order differential equations with anti-periodic boundary conditions: a survey. Bound. Value Probl. 2017(173), 1–27 (2017)

    MathSciNet  Google Scholar 

  20. Benchohra, M., Hamani, S., Ntouyas, S.K.: Boundary value problems for differential equations with fractional order and nonlocal conditions. Nonlinear Anal. 71, 2391–2396 (2009)

    MathSciNet  Google Scholar 

  21. Goodrich, C.: Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions. Comput. Math. Appl. 61, 191–202 (2011)

    MathSciNet  Google Scholar 

  22. Butt, R.L., Alzabut, J., Rehman, U.R., Jonnalagadda, J.: On fractional difference Langevin equations involving non-local boundary conditions. Dyn. Syst. Appl. 29(2), 305–326 (2020)

    Google Scholar 

  23. Seemab, A., Rehman, M., 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 

  24. Ulam, S.M.: A collection of the mathematical problems, pp. 665–666. Interscience, New York (1960)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  26. Obloza, M.: Hyers stability of the linear differential equation. Rocznik Naukowo-Dydaktyczny Prace Matematyczne 13, 259–270 (1993)

    MathSciNet  Google Scholar 

  27. Obloza, M.: Connections between Hyers and Lyapunov stability of the ordinary differential equations. Rocznik Naukowo-Dydaktyczny. Prace Matematyczne 14, 141–146 (1997)

    MathSciNet  Google Scholar 

  28. 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. 42(5), 2681–2699 (2019)

    MathSciNet  Google Scholar 

  29. 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)

    Google Scholar 

  30. Riaz, U., Zada, A., Ali, Z., et al.: Analysis of nonlinear coupled systems of impulsive fractional differential equations with Hadamard derivatives. Math. Probl. Eng. 2019, 1–20 (2019)

    MathSciNet  Google Scholar 

  31. Zada, A., Riaz, U., Khan, F.: Hyers-Ulam stability of impulsive integral equations. Boll. Unione Mat. Ital. 12(3), 453–467 (2019)

    MathSciNet  Google Scholar 

  32. Ntouyas, S.K., Tariboon, J., Sudsutad, W.: Boundary value problems for Riemann–Liouville fractional differential inclusions with nonlocal Hadamard fractional integral conditions. Mediter. J. Math. 13, 939–954 (2016)

    MathSciNet  Google Scholar 

  33. Maraaba, T.A., Jarad, F., Baleanu, D.: On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives. Sci. China Math. 51, 1775–1786 (2008)

    MathSciNet  Google Scholar 

  34. Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer Simul. 44, 460–481 (2017)

    MathSciNet  Google Scholar 

  35. Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012(1), 1–8 (2012)

    MathSciNet  Google Scholar 

  36. Cichon, M., Salem, H.A.H.: On the lack of equivalence between differential and integral forms of the Caputo-type fractional problems. J. Pseudo Differ. Oper. Appl. 11, 1869–1895 (2020)

    MathSciNet  Google Scholar 

  37. Salem, H.A.H., Cichon, M.: Analysis of tempered fractional calculus in Holder and Orlicz spaces. Symmetry 14, 1581 (2022)

    Google Scholar 

  38. Khan, Z.A., Shah, K., Abdalla, B., Abdeljawad, T.: A numerical study of complex dynamics of a chemostat model under fractal-fractional derivative. Fractals 31(8), 2340181 (2023)

    Google Scholar 

  39. Ahmed, S., Shah, K., Jahan, S., Abdeljawad, T.: An efficient method for the fractional electric circuits based on Fibonacci wavelet. Results Phys. 52, 106753 (2023)

    Google Scholar 

  40. Shah, K., Abdalla, B., Abdeljawad, T., Gul, R.: Analysis of multipoint impulsive problem of fractional-order differential equations. Bound. Value Probl. 2023(1), 1–17 (2023)

    MathSciNet  Google Scholar 

  41. Shah, K., Ali, G., Ansari, K.J., Abdeljawad, T., Meganathan, M., Abdalla, B.: On qualitative analysis of boundary value problem of variable order fractional delay differential equations. Bound. Value Probl. 2023(1), 1–15 (2023)

    MathSciNet  Google Scholar 

  42. Acay, B., Bas, E., Abdeljawad, T.: Non-local fractional calculus from different viewpoint generated by truncated M-derivative. J. Comput. Appl. Math. 366, 112410 (2020)

    MathSciNet  Google Scholar 

  43. Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012, 1–8 (2012)

    MathSciNet  Google Scholar 

  44. Adjabi, Y., Jarad, F., Baleanu, D., Abdeljawad, T.: On Cauchy problems with Caputo Hadamard fractional derivatives. J. Comput. Anal. Appl. 21(4), 1–21 (2016)

    MathSciNet  Google Scholar 

  45. Bedi, P., Kumar, A., Abdeljawad, T., Khan, A., Gomez-Aguilar, J.F.: Mild solutions of coupled hybrid fractional order system with Caputo–Hadamard derivatives. Fractals 29(6), 2150158 (2021)

    Google Scholar 

  46. Tajadodi, H., Khan, A., Gomez-Aguilar, J.F., Khan, H.: Optimal control problems with Atangana–Baleanu fractional derivative. Optim. Control Appl. Methods 42(1), 96–109 (2021)

    MathSciNet  Google Scholar 

  47. Khan, H., Alzabut, J., Baleanu, D., Alobaidi, G., Rehman, M.: Existence of solutions and a numerical scheme for a generalized hybrid class of n-coupled modified ABC-fractional differential equations with an application. AIMS Math. 8(3), 6609–6625 (2023)

    MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by Anhui Province Natural Science Research Foundation (2023AH051810).

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Authors and Affiliations

  1. Department of Common Course, Anhui Xinhua University, Hefei, 230088, Anhui, China

    Yanli Ma

  2. Department of Mathematics, University of Peshawar, Peshawar, Khyber Pakhtunkhwa, Pakistan

    Maryam Maryam & Akbar Zada

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

    Usman Riaz

  4. Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510 0 09, Alba Iulia, Romania

    Ioan-Lucian Popa

  5. Faculty of Mathematics and Computer Science, Transilvania University of Brasov, Iuliu Maniu Street 50, 500091, Brasov, Romania

    Ioan-Lucian Popa

  6. Mathematics Department, University of Prince Mugrin, P.O. Box 41040, 42241, Al Madinah, Saudi Arabia

    Lakhdar Ragoub

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  1. Yanli Ma
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  2. Maryam Maryam
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All authors contributed equally and significantly to this paper. All authors have read and approved the final version of the manuscript.

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Correspondence to Usman Riaz.

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Ma, Y., Maryam, M., Riaz, U. et al. Existence and Hyers–Ulam Stability of Jerk-Type Caputo and Hadamard Mixed Fractional Differential Equations. Qual. Theory Dyn. Syst. 23, 132 (2024). https://doi.org/10.1007/s12346-024-00971-8

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