A Class of Implicit Fractional $\psi$-Hilfer Langevin Equation with Time Delay and Impulse in the Weighted Space

Year 2024, Volume: 7 Issue: 2, 88 - 103, 30.06.2024

Ayoub Louakar , Ahmed Kajounı , Khalid Hilal , Hamid Lmou

https://doi.org/10.33434/cams.1425019

Cited By: 1

https://izlik.org/JA49XH36BD

Abstract

In this paper, the Ulam-Hyers-Rassias stability is discussed and the existence and uniqueness of solutions for a class of implicit fractional $\psi$-Hilfer Langevin equation with impulse and time delay are investigated. A novel form of generalized Gronwall inequality is introduced. Picard operator theory is employed in authour’s analysis. An example will be given to support the validity of our findings.

Keywords

Existence and uniqueness , Generalized Gronwall inequality , $\psi$-Hilfer Langevin equation , Picard operator theory , Ulam-Hyers-Rassias stability

Ethical Statement

There are no conflicts of interest, according to the authors.

Supporting Institution

No funding supporting.

Project Number

1

Thanks

The authors appreciate the referee's thoughtful comments on the manuscript, which helped to improve it.

References

• [1] J. V. D. C. Sousa, E. C. Capelas de Oliveira, On the y-Hilfer fractional derivative, Commun. Nonlinear Sci. Numer. Simul., 60 (2018), 72–91.
• [2] A. A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science Limited, 204 (2006), 7–10.
• [3] I. Podlubny, Fractional Differential equation, Academic Press, San Diego, 1999.
• [4] C. Beck, G. Roepstorff, From dynamical systems to the Langevin equation, Phys. A, 145(1-2) (1987), 1-14.
• [5] B. Ahmad, J. J. Nieto, A. Alsaedi, M. El-Shahed, A study of nonlinear Langevin equation involving two fractional orders in different intervals, Nonlinear Anal. Real World Appl., 13(2) (2012), 599–606.
• [6] S. Harikrishnan, K. Kanagarajan, E. M. Elsayed, Existence and stability results for Langevin equations with Hilfer fractional derivative, Res. Fixed Point Theory Appl., 20183 (2018).
• [7] K. Hilal, A. Kajouni, H. Lmou, Boundary value problem for the Langevin equation and inclusion with the Hilfer fractional derivative, Int. J. Differ. Equ., 2022 (2022) 1–12.
• [8] M. Aydin, N. I. Mahmudov, y-Caputo type time-delay Langevin equations with two general fractional orders, Math. Methods Appl. Sci., 46(8) (2023), 9187-9204.
• [9] I. T. Huseynov, N. I. Mahmudov, A class of Langevin time-delay differential equations with general fractional orders and their applications to vibration theory, J. King Saud Univ. Sci., 33(8) (2021), 101596.
• [10] M. Aydin, Langevin delayed equations with Prabhakar derivatives involving two generalized fractional distinct orders, Turkish J. Math., 48 (2024), 144-162
• [11] N. I. Mahmudov, A. Ahmadova, I. T. Huseynov, A novel technique for solving Sobolev type fractional multi-order evolution equations, Comput. Appl. Math., 41(2) (2022), 1–35.
• [12] M. Aydin, N. I. Mahmudov, Some applications of the generalized Laplace transform and the representation of a solution to Sobolev-type evolution equations with the generalized Caputo derivative, Bull. Polish Acad. Sci. Tech. Sci., 72(2) (2024).
• [13] M. Aydin, N. I. Mahmudov, The Sequential Conformable Langevin-Type Differential Equations and Their Applications to the RLC Electric Circuit Problems, J. Appl. Math., 2024 (2024), 1–14.
• [14] Z. Shi, Y. Li, H. Cheng, Dynamic analysis of a pest management smith model with impulsive state feedback control and continuous delay, Mathematics, 7(7) (2019), 591.
• [15] U. Forys, J. Poleszczuk, T. Liu, Logistic tumor growth with delay and impulsive treatment, Math. Population Stud., 21 (2014), 146–158.
• [16] S. M. Ulam, A collection of mathematical problems, Interscience Publishers, 1960.
• [17] T. M. Rassias, On the stability of linear mappings in Banach spaces, Proc. Amer. Math. Soc., 72 (1978), 297–300.
• [18] R. Rizwan, A. Zada, H. Waheed, U. Riaz, Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives, Int. J. Nonlin. Sci. Num., 24(6) (2023), 2405–2423.
• [19] M. S. Abdo, S. K. Panchal, H.A. Wahash, Ulam–Hyers–Mittag-Leffler stability for a y-Hilfer problem with fractional order and infinite delay, Results Appl. Math., 7(100) (2020), 115.
• [20] K. B. Lima, J. V. D. C. Sousa, E. C. Capelas de Oliveira, Ulam–Hyers type stability for y-Hilfer fractional differential equations with impulses and delay, Comput. Appl. Math., 40(293) (2021).
• [21] J. V. C. Sousa, E.C. Capelas de Oliveira, A Gronwall inequality and the Cauchy type problem by means of Hilfer operator, Diff. Equ. and Appl., 11(1) (2019) 87-106.
• [22] R. Rizwan, J. R. Lee, C. Park, A. Zada, Existence, uniqueness and Ulam’s stabilities for a class of impulsive Langevin equation with Hilfer fractional derivatives, AIMS Mathematics, 7(4) (2022), 6204–6217
• [23] Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, 2014.
• [24] I. A. Rus, Gronwall lemmas: Ten open problems, Sci. Math. Jpn., 70 (2019), 221-228.
• [25] J. Alzabut, Y. Adjabi, W. Sudsutad, M. ur Rehman, New generalizations for Gronwall type inequalities involving a y-fractional operator and their applications, AIMS Math., 6 (2021), 5053–5077.
• [26] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 94 (2012). https://doi.org/10.1186/1687-1812-2012-94.