Home Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative
Article
Licensed
Unlicensed Requires Authentication

Existence results for a coupled system of fractional stochastic differential equations involving Hilfer derivative

  • Fatima Zahra Arioui EMAIL logo
Published/Copyright: August 3, 2024

Abstract

In this paper, we consider a coupled system of fractional stochastic differential equations involving the Hilfer derivative of order 1 2 < α < 1 . Under some assumptions, we prove the existence of mild solutions for our system based on Perov’s and Schaefer’s fixed point theorems. An example illustrating our result is provided.

MSC 2020: 60H10; 26D10

Communicated by Nikolai Leonenko


References

[1] S. Abbas, B. Ahmad, M. Benchohra and A. Salim, Fractional Difference, Differential Equations, and Inclusions: Analysis and Stability, Elsevier, Amsterdam, 2024. Search in Google Scholar

[2] R. Agarwal, S. Hristova and D. O’Regan, Non-instantaneous impulses in Caputo fractional differential equations, Fract. Calc. Appl. Anal. 20 (2017), no. 3, 595–622. 10.1515/fca-2017-0032Search in Google Scholar

[3] R. Agarwal, S. Hristova and D. O’Regan, Existence and integral representation of scalar Riemann–Liouville fractional differential equations with delay and impulses, Mathematics 8 (2020), 10.3390/math8040607. 10.3390/math8040607Search in Google Scholar

[4] H. M. Ahmed and M. M. El-Borai, Hilfer fractional stochastic integro-differential equations, Appl. Math. Comput. 331 (2018), 182–189. 10.1016/j.amc.2018.03.009Search in Google Scholar

[5] H. M. Ahmed, M. M. El-Borai and M. E. Ramadan, Noninstantaneous impulsive and nonlocal Hilfer fractional stochastic integrodifferential equations with fractional Brownian motion and Poisson jumps, Int. J. Nonlinear Sci. Numer. Simul. 22 (2021), no. 7–8, 927–942. 10.1515/ijnsns-2019-0274Search in Google Scholar

[6] F. Z. Arioui, Weighted fractional stochastic integro-differential equation with infinite delay, Arab. J. Math. 12 (2023), no. 3, 499–511. 10.1007/s40065-023-00430-3Search in Google Scholar

[7] A. Boutiara, J. Alzabut, M. Ghaderi and S. Rezapour, On a coupled system of fractional ( p , q ) -differential equation with Lipschitzian matrix in generalized metric space, AIMS Math. 8 (2023), no. 1, 1566–1591. 10.3934/math.2023079Search in Google Scholar

[8] R. Chaudhary, M. Muslim and D. N. Pandey, Approximation of solutions to fractional stochastic integro-differential equations of order α ( 1 , 2 ] , Stochastics 92 (2020), no. 3, 397–417. 10.1080/17442508.2019.1625904Search in Google Scholar

[9] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University, Cambridge, 1992. 10.1017/CBO9780511666223Search in Google Scholar

[10] K. Dhawan, R. K. Vats and R. P. Agarwal, Qualitative analysis of coupled fractional differential equations involving Hilfer derivative, An. Ştiinţ. Univ. “Ovidius” Constanţa Ser. Mat. 30 (2022), no. 1, 191–217. 10.2478/auom-2022-0011Search in Google Scholar

[11] K. Diethelm, The Analysis of Fractional Differential Equations, Lecture Notes in Math. 2004, Springer, Berlin, 2010. 10.1007/978-3-642-14574-2Search in Google Scholar

[12] J. Du, W. Jiang and A. U. K. Niazi, Approximate controllability of impulsive Hilfer fractional differential inclusions, J. Nonlinear Sci. Appl. 10 (2017), no. 2, 595–611. 10.22436/jnsa.010.02.23Search in Google Scholar

[13] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 1998. Search in Google Scholar

[14] K. M. Furati, M. D. Kassim and N. Tatar, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl. 64 (2012), no. 6, 1616–1626. 10.1016/j.camwa.2012.01.009Search in Google Scholar

[15] H. Gou, Y. Li and Q. Li, Mixed monotone iterative technique for Hilfer fractional evolution equations with nonlocal conditions, J. Appl. Anal. Comput. 10 (2020), no. 5, 1823–1847. 10.11948/20190211Search in Google Scholar

[16] H. Gu and J. J. Trujillo, Existence of mild solution for evolution equation with Hilfer fractional derivative, Appl. Math. Comput. 257 (2015), 344–354. 10.1016/j.amc.2014.10.083Search in Google Scholar

[17] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, 2000. 10.1142/9789812817747Search in Google Scholar

[18] R. Hilfer, Y. Luchko and Ž. Tomovski, Operational method for the solution of fractional differential equations with generalized Riemann–Liouville fractional derivatives, Fract. Calc. Appl. Anal. 12 (2009), no. 3, 299–318. Search in Google Scholar

[19] P. Kalamani, D. Baleanu, S. Selvarasu and M. M. Arjunan, On existence results for impulsive fractional neutral stochastic integro-differential equations with nonlocal and state-dependent delay conditions, Adv. Difference Equ. 2016 (2016), Paper No. 163. 10.1186/s13662-016-0885-4Search in Google Scholar

[20] R. Kamocki, A new representation formula for the Hilfer fractional derivative and its application, J. Comput. Appl. Math. 308 (2016), 39–45. 10.1016/j.cam.2016.05.014Search in Google Scholar

[21] A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud. 204, Elsevier Science, Amsterdam, 2006. Search in Google Scholar

[22] J. Lv and X. Yang, A class of Hilfer fractional stochastic differential equations and optimal controls, Adv. Difference Equ. 2019 (2019), Paper No. 17. 10.1186/s13662-019-1953-3Search in Google Scholar

[23] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Približ. Metod. Rešen. Differ. Uravnen. (1964), no. 2, 115–134. Search in Google Scholar

[24] I. Podlubny, Fractional Differential Equations, Math. Sci. Eng. 198, Academic Press, San Diego, 1999. Search in Google Scholar

[25] Y. Qin, Analytic Inequalities and Their Applications in PDEs, Oper. Theory Adv. Appl. 241, Birkhäuser/Springer, Cham, 2017. 10.1007/978-3-319-00831-8Search in Google Scholar

[26] S. Sivasankar, R. Udhayakumar, M. Hari Kishor, S. Alhazmi and S. Al-Omari, A new result concerning nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators, Mathematics 11 (2023), 10.3390/math11010159. 10.3390/math11010159Search in Google Scholar

[27] S. Sivasankar, R. Udhayakumar, V. Subramanian, G. AlNemer and A. Elshenhab, Existence of Hilfer fractional stochastic differential equations with nonlocal conditions and delay via almost sectorial operators, Mathematics 10 (2022), 10.3390/math10224392. 10.3390/math10224392Search in Google Scholar

[28] H. A. Wahash, M. S. Abdo, S. K. Panchal and S. P. Bhairat, Existence of solution for Hilfer fractional differential problem with nonlocal boundary condition in Banach spaces, Stud. Univ. Babeş-Bolyai Math. 66 (2021), no. 3, 521–536. 10.24193/subbmath.2021.3.09Search in Google Scholar

[29] J. Wang, A. G. Ibrahim and D. O’Regan, Finite approximate controllability of Hilfer fractional semilinear differential equations, Miskolc Math. Notes 21 (2020), no. 1, 489–507. 10.18514/MMN.2020.2921Search in Google Scholar

Received: 2023-10-24
Accepted: 2024-04-05
Published Online: 2024-08-03
Published in Print: 2024-11-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 29.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/rose-2024-2015/html
Scroll to top button