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Existence of mild solution for fuzzy fractional differential equation utilizing the Hilfer-Katugampola . . .
The linear operators A : Dom(A) ⊆ B → B is Author contributions
given by
Conceptualization: All authors
Formal analysis: Ramaraj Hariharan
2
∂ ex
D(A) = e x ∈ H : ∈ B and Investigation: Ramalingam Udhayakumar
∂x 2 Methodology: Ramaraj Hariharan

Project administration: Ramalingam Udhayaku-
e x(0, 0) = ex(0, 1) = 0 ,
mar
Resources: Ramaraj Hariharan
and
2
∂ ex Supervision: Ramalingam Udhayakumar
Aex = . Validation: All authors
∂ℓ 2
Visualization: Ramalingam Udhayakumar
Then, we get Writing – original draft: Ramaraj Hariharan
Writing – review & editing: All authors
D(A) = {ex ∈ H : ex(t, 0) = ex(t, 1) = 0}.

Availability of data
Therefore, A is bounded and it forms a compact
C 0 -semigroup {T(t)} t≥0 on D(A). Not applicable.
Consequently, system (7) can be rewritten as sys-
tem (1). The function F satisfies the assumptions
(H1)−(H4). This implies that Theorem 3 is fully References
satisfied. Therefore, there is a mild solution for [1] Zhou, Y. (2023). Basic theory of fractional differ-
fuzzy fractional differential equation (1) on U 1 . ential equations. World scientific.
[2] Ma, Y. K., Dineshkumar, C., Vijayakumar, V.,
Udhayakumar, R., Shukla, A., & Nisar, K. S.
5. Conclusion
(2023). Approximate controllability of Atangana-
Baleanu fractional neutral delay integrodifferen-
In this paper we have investigated the existence
tial stochastic systems with nonlocal conditions.
of mild solutions for fuzzy fractional differen-
Ain shams Engineering journal, 14(3), 101882. ht
tial equations using the Hilfer-Katugampola frac- tps://doi.org/10.1016/j.asej.2022.101882
tional derivative, which is a generalization of the [3] Nandhaprasadh, K., & Udhayakumar, R. (2024).
Riemann-Liouville and Hadamard derivative. We Hilfer Fractional Neutral Stochastic Differential
establish the necessary conditions for these so- Inclusions with Clarke’s Subdifferential Type and
lutions by applying Schauder’s fixed point the- fBm: Approximate Boundary Controllability.
orem, a key tool in functional analysis. To make Contemporary Mathematics, 1013-1035. https:
the findings more practical, we provide a detailed //doi.org/10.37256/cm.5120243580
example demonstrating how the theoretical re- [4] Varun Bose, C. S., & Udhayakumar, R. (2022).
A note on the existence of Hilfer fractional differ-
sults can be applied. Looking ahead, future re-
ential inclusions with almost sectorial operators.
search will focus on exploring the controllability
Mathematical Methods in the Applied Sciences,
and stability of these solutions under varying con-
45(5), 2530-2541. https://doi.org/10.1002/
ditions, broadening their applicability in complex
mma.7938
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maran, V., Gokul, G., & Al-Omari, S. (2024).
Existence of Hilfer fractional neutral stochastic
Acknowledgments
differential systems with infinite delay. Bulletin
of the Karaganda University. Mathematics series,
None.
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024m1/174-193
Funding [6] Kilbas, A. A. (2006). Theory and applications
of fractional differential equations. North-Holland
None. Mathematics Studies, 204.
[7] Miller, K.S., & Ross, B. (1993). An Introduction
to the Fractional Calculus and Differential Equa-
Conflict of interest tions. John Wiley, NewYork.
[8] Lakshmikantham, V., Leela, S., & Devi, J. V.
The authors declare that they have no conflict of (2009). Theory of Fractional Dynamic Systems,
interest regarding the publication of this article. Cambridge Scientific Publishers.
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