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Existence of mild solution for fuzzy fractional differential equation utilizing the Hilfer-Katugampola . . .
In 2011, Katugampola 11 introduced fractional op- framework, the study aims to provide accurate
erators named after him: the Katugampola frac- numerical solutions for problems involving frac-
tional integral and the Katugampola fractional tional derivatives and uncertainty.
derivative. These operators include an additional Arshad 19 delved into Fuzzy fractional differential
parameter, ϱ > 0. As approaches 0 + , they con-
equations properties, investigating their existence
verge to the Hadamard fractional operators, and
and uniqueness via fuzzy integral equivalent equa-
when ϱ = 1, they correspond to the Riemann- 20
tions. Allahviranloo et al. contributed to this
Liouville fractional operators. This dependence
field by examining Fuzzy fractional differential
on ϱ simplifies the theoretical framework because
equations properties under the Caputo fractional
proving a result for the Katugampola derivative
gH-derivative, particularly focusing on their exis-
simultaneously proves it for both the Riemann- 21
tence and uniqueness. Zhu and Rao considered
Liouville and Hadamard derivatives. These oper-
fuzzy differential inclusion in the Banach space U
ators are gaining popularity and are extensively
in 2000. Fuzzy differential inclusion can increase
discussed in current literature.
the uncertainty of the studied system, and is also
Hilfer’s 12 work on the Hilfer fractional deriva- an effective method to study the existing results.
tive, developed from the Riemann-Liouville and
Caputo derivatives, has sparked significant inter- 22
In, Chen, Gu, and wang considered the Hilfer-
est due to its wide-ranging applications. Gu and
Trujillo 13 investigated the existence of mild solu- Katugampola fractional differential equations
fuzzy set with the nonlocal condition:
tions to evolution equations involving the Hilfer
fractional derivative. Their research focuses on
( ϱ
a,b
extending the understanding of fractional calculus D + u (t) = f(t, u(t)), t ∈ [x, y]
x
and its applications to evolution equations, which ϱ 1−µ
I + u (0) = u 0 ,
are crucial in modeling various physical and en- x
gineering processes. The Hilfer fractional deriv-
where u ∈ R, a ∈ (0, 1), b ∈ [0, 1], µ = a+b(1−a)
ative, a generalization that bridges the gap be-
and ϱ > 0, f : [x, y] × X → X is a fuzzy func-
tween the Riemann-Liouville and Caputo deriva- ϱ 1−µ ϱ a,b
tives, allows for more flexible modeling of memory tion. furthermore, I + , D + are the Hilfer-
x
x
and hereditary properties in dynamical systems. Katugampola fractional integral and derivative.
Oliveira and Oliveira 14 introduced and explored Based on the discussion above, this paper explores
a new type of fractional derivative that combines the existence of mild solutions for fuzzy fractional
the properties of the Hilfer and Katugampola differential equations, incorporating the Hilfer-
fractional derivatives. This novel derivative pro- Katugampola fractional derivative, with the fol-
vides a versatile tool for modeling and analyzing lowing initial condition:
systems with complex, non-local dynamics. The
authors present the mathematical formulation of (
τ,ω
ϱ D + ex(t) = Aex(t) + F(t, ex(t)),
the Hilfer-Katugampola fractional derivative and 0 (1)
ϱ 1−µ
discuss its potential applications in various fields. I + ex(0) = ex 0 ,
0
ϱ
Bede and Stefanini, 15 explored the concept of dif- where D τ,ω is a Hilfer-Katugampola fractional
0 +
ferentiability for functions that take fuzzy values. derivative of order τ ∈ (0, 1) and ω ∈ [0, 1],
The authors extend the classical notions of dif- t ∈ [0, u] = U, µ = τ + ω(1 − τ) and ϱ > 0.
ferentiability to accommodate fuzzy-valued func- The state variable ex(·) takes on values within the
tions, providing a comprehensive framework for space of all fuzzy numbers H. A fuzzy number is
their analysis. Fuzzy numbers, initially intro- defined as a fuzzy set ex : R → [0, 1]. A is an in-
duced by Chang and Zadeh 16 in 1972, provide finitesimal generator of a compact operator semi-
valuable means for quantifying uncertainty within group {J(t)} t≥0 , F : U × B → H is a fuzzy func-
ϱ 1−µ
mathematical frameworks. The notion of fuzzy tion. Moreover, I + is the Hilfer-Katugampola
0
fractional differential equations was pioneered by fractional integral.
Agarwal et al. 17 and has since been a subject of
This article’s primary benefit and contribution
extensive exploration, covering theories, practi-
can be put up as follows:
cal applications, and solution methodologies. Ar-
qub et al. 18 investigated a novel computational (1) In this paper, we explore the existence
method to solve fuzzy fractional initial value prob- of a mild solution for fuzzy fractional
lems. By utilizing the Mittag–Leffler kernel dif- differential equation utilizing the Hilfer-
ferential operator within the reproducing kernel Katugampola fractional derivative.
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