Dynamics and Stability of $\Xi$-Hilfer Fractional Fuzzy Differential Equations with Impulses

Year 2023, Volume: 6 Issue: 3, 115 - 127, 17.09.2023

Ravichandran Vıvek Kangarajan K. , Dvivek Vivek , Elsayed Elsayed

https://doi.org/10.33434/cams.1257750

Cited By: 1

Abstract

This paper deals with the existence, uniqueness, and Ulam-stability outcomes for $\Xi$-Hilfer fractional fuzzy differential equations with impulse. Further, by using the techniques of nonlinear functional analysis, we study the Ulam-Hyers-Rassias stability.

Keywords

Existence, Fuzzy impulsive differential equation, $\Xi$-Hilfer fractional derivative, Uniqueness, Ulam-Hyers-Rassias stability

Supporting Institution

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Project Number

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Thanks

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References

• [1] M. Benchohra, J.J. Nieto, A. Ouahab, Fuzzy solutions for impulsive differential equations, Commun. Appl. Anal., 11(2007), 379-394.
• [2] M. Benchohra, J. Henderson, S.L. Ntouyas, Impulsive Differential Equations and Inclusions, New York, Hindawi Publishing Corporation, 2(2006).
• [3] M. Feckan, Y. Zhou, J. Wang, On the concept and existence of solutions for impulsive fractional differential equations, Commun. Nonlinear. Sci. Numer. Simul., 17(7)(2012), 3050-3060.
• [4] T.L. Guo, W. Jiang, Impulsive fractional functional differential equations, Comput. Math. Appl., 64(2012), 3414-3424.
• [5] N.V. Hoa, D. O’Regan, A remark on y-Hilfer fractional differential equations with non-instantaneous impulses, Math. Methods. Appl. Sci., 43(2020), 3354-3368.
• [6] N.V. Hoa, T.V. An, Fuzzy differential equations with Riemann-Liouville generalized fractional integrable impulses, Fuzzy Sets Syst., 2021.
• [7] D. Luo, Z. Luo, Existence and Hyers-Ulam stability results for a class of fractional order delay differential equations with non-instantaneous impulses, Math. Slovaca., 70(5)(2020), 1231-1248.
• [8] J.V.D.C. Sousa, K.D. Kucche, E.C. de Oliveira, Stability of y-Hilfer impulsive fractional differential equations, Appl. Math. Lett., 88(2019), 73-80.
• [9] J.V.D.C. Sousa, D.D.S. Oliveira, E.C. de Oliveira, On the existence and stability for non-instantaneous impulsive fractional integro-differential equations, Math. Methods. Appl. Sci., 42(2019), 1249-1261.
• [10] K.M. Furati, M.D. Kassim, Existence and uniqueness for a problem involving Hilfer fractional derivative, Comput. Math. Appl., 64(6)(2012), 1616-1626.
• [11] R. Hilfer, Applications of Fractional Calculus in Physics, Singapore, World scientific, 2000.
• [12] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
• [13] I. Podlubny, Fractional Differential Equation, Academic Press, San Diego, 1999.
• [14] J.V.D.C. Sousa, E.C. de Oliveira, On the y-Hilfer fractional derivative, Commun. Nonlinear. Sci. Numer. Simul., 60(2018), 72-91.
• [15] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy sets syst., 230(2013), 119-141.
• [16] V. Lakshikantham, R.N. Mohapatra, Theory of fuzzy differential equations and applications, London, CRC Press, 2003.
• [17] L. Sajedi, N. Eghbali, H. Aydi, Impulsive coupled system of fractional differential equations with Caputo-Katugampola fuzzy fractional derivative, Hindawi. J. Math., 13(2021).
• [18] X. Chen, H. Gu, X. Wang, Existence and uniqueness for fuzzy differential equation with Hilfer-Katugampola fractional derivative, Adv. Differ. Equ., 2020(2020), 241.
• [19] N.V. Hoa, H. Vu, T.M. Duc, Fuzzy fractional differential equations under Caputo-Katugampola fractional derivative approach, Fuzzy Sets Syst., 375(2019), 70-99.
• [20] N.V. Hoa, Fuzzy fractional functional differential equations under Caputo gH-differentiability, Commun. Nonlinear. Sci. Numer. Simul., 22(1-3)(2015), 1134-1157.
• [21] D.F. Luo, T. Abdeljawad, Z.G. Luo, Ulam-Hyers stability results for a novel nonlinear nabla caputo fractional variable order difference system, Turk. J. Math., 45(1)(2021), 456-70.
• [22] D. Luo, K. Shah, Z. Luo, On the novel Ulam-Hyers stability for a class of nonlinear y-Hilfer fractional differential equation with time-varying delays, Mediterr. J. Math., 16(5)(2019), 112.
• [23] S. Rashid, F. Jarad, K.M. Abualnaja, On fuzzy Volterra-Fredholm integrodifferential equation associated with Hilfergeneralized proportional fractional derivative, AIMS Math., 6(2021), 10920-10946.
• [24] H. Vu, J.M. Rassias, N.V. Hoa, Ulam-Hyers-Rassias stability for fuzzy fractional integral equations, Iran. J. Fuzzy syst., 17(2020), 17-27.
• [25] X. Wang, D. Luo, Q. Zhu, Ulam-Hyers stability of Caputo type fuzzy fractional differential equation with time-delays, Chaos. Solitons. Fractals., 156(2022), 111822.