New results for fractional ordinary differential equations in fuzzy metric space

Li Chen; Suyun Wang; Yongjun Li; Jinying Wei; Li Chen; Suyun Wang; Yongjun Li; Jinying Wei

doi:10.3934/math.2024674

AIMS Mathematics

2024, Volume 9, Issue 6: 13861-13873. doi: 10.3934/math.2024674

Previous Article Next Article

Research article Special Issues

New results for fractional ordinary differential equations in fuzzy metric space

School of Information Engineering, Lanzhou City University, Lanzhou, 730070, China

Received: 17 January 2024 Revised: 27 March 2024 Accepted: 08 April 2024 Published: 15 April 2024
MSC : 34A08, 34B15, 35J05

In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.
- fuzzy metric space,
- fractional fuzzy differential equation,
- power compression mapping principle,
- existence and uniqueness
Citation: Li Chen, Suyun Wang, Yongjun Li, Jinying Wei. New results for fractional ordinary differential equations in fuzzy metric space[J]. AIMS Mathematics, 2024, 9(6): 13861-13873. doi: 10.3934/math.2024674

Related Papers:

Abstract

In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.

References

[1]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, New York: Wiley, 1993.
[2]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Amsterdam, Boston: Elsevier, 2006.
[3]	N. A. Albasheir, A. Alsinai, A. U. K. Niazi, R. Shafqat, Romana, M. Alhagyan, et al., A theoretical investigation of Caputo variable order fractional differential equations: Existence, uniqueness, and stability analysis, Comput. Appl. Math., 42 (2023), 367. https://doi.org/10.1007/S40314-023-02520-6 doi: 10.1007/S40314-023-02520-6
[4]	V. Lakshmikantham, A. S. Vatsala, Basic theory of fractional differential equations, Nonlinear Anal., 69 (2008), 2677–2682. https://doi.org/10.1016/j.na.2007.08.042 doi: 10.1016/j.na.2007.08.042
[5]	M. Al-Refai, D. Baleanu, Comparison principles of fractional differential equations with non-local derivative and their applications, AIMS Mathematics, 6 (2021), 1443–1451. https://doi.org/10.3934/math.2021088 doi: 10.3934/math.2021088
[6]	L. Chen, G. Duan, S. Y. Wang, J. F. Ma, A Choquet integral based fuzzy logic approach to solve uncertain multi-criteria decision making problem, Expert Syst. Appl., 149 (2020), 113303. https://doi.org/10.1016/j.eswa.2020.113303 doi: 10.1016/j.eswa.2020.113303
[7]	M. H. Alshayeji, S. C. Sindhu, S. Abed, Viral genome prediction from raw human DNA sequence samples by combining natural language processing and machine learning techniques, Expert Syst. Appl., 218 (2023), 119641. https://doi.org/10.1016/j.eswa.2023.119641 doi: 10.1016/j.eswa.2023.119641
[8]	K. Tamilselvan, V. Visalakshi, P. Balaji, Applications of picture fuzzy filters: Performance evaluation of an employee using clustering algorithm, AIMS Mathematics, 8 (2023), 21069–21088. https://doi.org/10.3934/math.20231073 doi: 10.3934/math.20231073
[9]	S. Salahshour, A. Ahmadian, B. A. Pansera, M. Ferrara, Uncertain inverse problem for fractional dynamical systems using perturbed collage theorem, Commun. Nonlinear Sci. Numer. Simul., 94 (2021), 105553. https://doi.org/10.1016/j.cnsns.2020.105553 doi: 10.1016/j.cnsns.2020.105553
[10]	M. Z. Ahmad, M. K. Hasan, B. D. Baets, Analytical and numerical solutions of fuzzy differential equations, Inf. Sci., 236 (2013), 156–167. https://doi.org/10.1016/j.ins.2013.02.026 doi: 10.1016/j.ins.2013.02.026
[11]	M. Mazandarani, N. Pariz, A. V. Kamyad, Granular differentiability of fuzzy-number-valued functions, IEEE Trans. Fuzzy Syst., 26 (2018), 310–323. https://doi.org/10.1109/TFUZZ.2017.2659731 doi: 10.1109/TFUZZ.2017.2659731
[12]	Y. B. Shao, Q. Mou, Z. T. Gong, On retarded fuzzy functional differential equations and nonabsolute fuzzy integrals, Fuzzy Sets Syst., 375 (2019), 121–140. https://doi.org/10.1016/j.fss.2019.02.005 doi: 10.1016/j.fss.2019.02.005
[13]	T. Allahviranloo, A. Armand, Z. Gouyandeh, Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, J. Intell. Fuzzy Syst., 26 (2014), 1481–1490.
[14]	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. https://doi.org/10.1016/j.fss.2018.08.001 doi: 10.1016/j.fss.2018.08.001
[15]	M. Mazandarani, A. V. Kamyad, Modified fractional Euler method for solving fuzzy fractional intial value problem, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), 12–21. https://doi.org/10.1016/j.cnsns.2012.06.008 doi: 10.1016/j.cnsns.2012.06.008
[16]	C. X. Wu, Z. T. Gong, On Henstock integrals of interval-valued and fuzzy-number-valued functions, Fuzzy Sets Syst., 115 (2000), 377–391. https://doi.org/10.1016/S0165-0114(98)00277-2 doi: 10.1016/S0165-0114(98)00277-2
[17]	D. Denneberg, Non-additive measure and integral, Dordrecht: Springer, 1994. https://doi.org/10.1007/978-94-017-2434-0
[18]	Z. T. Gong, L. Chen, G. Duan, Choquet integral of fuzzy-number-valued functions: The differentiability of the primitive with respect to fuzzy measures and choquet integral equations, Abstr. Appl. Anal., 2014 (2014), 953893. https://doi.org/10.1155/2014/953893 doi: 10.1155/2014/953893
[19]	C. V. Negoiţă, D. A. Ralescu, Application of fuzzy sets to system analysis, Basel: Springer, 1975. https://doi.org/10.1007/978-3-0348-5921-9
[20]	L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets Syst., 161 (2010), 1564–1584. https://doi.org/10.1016/j.fss.2009.06.009 doi: 10.1016/j.fss.2009.06.009
[21]	P. Diamond, P. Kloeden, Metric topology of fuzzy numbers and fuzzy analysis, In: Fundamentals of fuzzy sets, Boston: Springer, 2000. https://doi.org/10.1007/978-1-4615-4429-6_12
[22]	B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets Syst., 230 (2013), 119–141. https://doi.org/10.1016/j.fss.2012.10.003 doi: 10.1016/j.fss.2012.10.003
[23]	S. Salahshour, T. Allahviranloo, S. Abbasbandy, D. Baleanu, Existence and uniqueness results for fractional differential equations with uncertainty, Adv. Differ. Equ., 2012 (2012), 112. https://doi.org/10.1186/1687-1847-2012-112 doi: 10.1186/1687-1847-2012-112

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)