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Discontinuity, Nonlinearity, and Complexity 14(2) (2025) 407--415 | DOI:10.5890/DNC.2025.06.012

Abdulrahman A. Sharif$^{1,2}$, Maha M. Hamood$^{2,3}$, Ahmed A. Hamoud$^{3}$, Kirtiwant P. Ghadle$^2$

$^{1}$ Department of Mathematics, Hodeidah University, AL-Hudaydah, Yemen

$^{2}$ Department of Mathematics, Dr. Babasaheb Ambedkar Marathwada University, Aurangabad, India

$^{3}$ Department of Mathematics, Taiz University, Taiz P.O. Box 6803, Yemen

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Abstract

In this paper, we aim to prove new results about the existence and uniqueness of solutions to fuzzy fractional Volterra-Fredholm integro-differential equations (FFV-FIDEs). These equations include fuzzy beginning conditions and generalized fuzzy Caputo-Hukuhara differentiability. The consecutive iteration approach and the Banach fixed point theorem are used in the proof. An example is provided to illustrate the main results.

References

1. [1]	Agarwal, R.P., Lakshmikantham, V., and Nieto, J.J. (2010), On the concept of solution for fractional differential equations with uncertainty, Nonlinear Analysis: Theory, Methods $\&$ Applications, 72, 59-62.

2. [2]	Sharif, A.A., Hamood, M.M., and Khandagaale, A.D. (2022), Usage of the fuzzy Laplace transform method for solving one-dimensional fuzzy integral equations, Equations, 2, 2732-9976.

3. [3]	Podlubny, I. (1999), Fractional Differential Equations, Mathematics in Science and Engineering, 198, Academic Press.

4. [4]	Dawood, L., Sharif, A., and Hamood, A. (2020), Solving higher-order integro-differential equations by VIM and MHPM, International Journal of Applied Mathematics, 33(2), 253-264.

5. [5]	Hamoud, A. and Sharif, A. (2023), Existence, uniqueness and stability results for nonlinear neutral fractional Volterra-Fredholm integro-differential equations, Discontinuity, Nonlinearity, and Complexity, 12(2), 381-398.

6. [6]	Armand, A. and Gouyandeh, Z. (2015), Fuzzy fractional integro-differential equations under generalized Caputo differentiability, Annals of Fuzzy Mathematics and Informatics, 10(5), 789-798.

7. [7]	Sharif, A. and Hamoud A. (2022), On $\psi-$Caputo fractional nonlinear Volterra-Fredholm integro-differential equations, Discontinuity, Nonlinearity, and Complexity, 11(1), 97-106

8. [8]	Bede, B. and Gal, S.G. (2005), Generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151, 581-599.

9. [9]	Salahshour, S., Allahviranloo, T., and Abbasbandy, S. (2012), Solving fuzzy fractional differential equations by fuzzy Laplace transforms, Communications in Nonlinear Science and Numerical Simulation, 17, 1372-1381.

10. [10]	Sharif A., Hamoud, A., and Ghadle, K. (2023), On existence and uniqueness of solutions to a class of fractional Volterra-Fredholm initial value problems, Discontinuity, Nonlinearity, and Complexity, 12(4), 905-916.

11. [11]	Al-Smadi, M., Arqub, O., and Zeidan, D. (2021), Fuzzy fractional differential equations under the Mittag-Leffler kernel differential operator of the ABC approach: Theorems and applications, Chaos, Solitons $\&$ Fractals, 146, 110891.

12. [12]	Bede, B. and Stefanini, L. (2005), Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230, 119-141.

13. [13]	Allahviranloo, T., Gouyandeh, Z., and Armand, A. (2000), Fuzzy fractional differential equations under generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems, 26(3), 1481-1490. DOI: 10.3233/IFS130831.

14. [14]	Arshad, S. and Luplescu, V. (2011), On the fractional differential equations with uncertainty, Nonlinear Analysis, 74, 85-93.

15. [15]	Arshad, S. and Luplescu, V. (2011), Fractional differential equation with fuzzy initial condition, Electronic Journal of Differential Equations, 34, 1-8.

16. [16]	Allahviranloo, T., Abbasbandy, S., and Salahshour, S. (2011), Fuzzy Fractional Differential Equations with Nagumo and Krasnoselskii-Krein Condition, In: Eusflat-LFA, Aix-les-Bains, France.

17. [17]	Agilan, K. and Parthiban, V. (2022), Existence results for fuzzy fractional Volterra integro differential equations, International Conference on Mathematical Techniques and Applications, 18, 2516, 180002.

18. [18]	Anastassiou, G.A. (2010), Fuzzy Mathematics: Approximation Theory, Studies in Fuzziness and Soft Computing, Springer-Verlag Berlin Heidelberg, 251, 1434-9922.

19. [19]	Esfahani, A., Solaymani, O., and Aliabdoli, T. (2014), On the existence and uniqueness of solutions to fuzzy boundary value problems, Annals of Fuzzy Mathematics and Informatics, 7, 15-29.

20. [20]	Stefanini, L. and Bede, B. (2009), Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis, 71, 1311-1328.

21. [21]	Goetschel, R. and Voxman, W. (1986), Elementary calculus, Fuzzy Sets Systems, 18, 31-43.

22. [22]	Lakshmikantham, V. and Mohapatra, R. (2003), Theory of Fuzzy Differential Equations and Inclusions, Taylor and Francis, London.

23. [23]	Lakshmikantham, V., Gnana, B., and Vasundhara, D. (2006), Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific Publishers.