Page 90 - IJOCTA-15-1

R. Hariharan, R. Udhayakumar / IJOCTA, Vol.15, No.1, pp.82-91 (2025)
(2) A mild solution to system (1) is con- Remark 1. Given that ex ∈ R, ex is regarded as a
structed using the Laplace transform. specific point in F. We define Com c (R) as the set
(3) Using Schauder’s fixed point theorem, we of all nonempty, compact, convex subsets of R.
proved the existence of a fixed point for a For any elements ex i , ex j ∈ Com c (R), the Haus-
mild solution. dorff distance between ex i and ex j is defined as:
(4) A theoretical example is presented to il-
lustrate the proposed results.
d(ex i , ex j ) = max sup inf ∥f − g∥,
Additionally, the remainder of the paper has been g∈ex j
f∈ex i
categorized as follows: We get the fundamental
ideas of fuzzy fractional calculus relevant to our sup inf ∥f − g∥ .
investigation from Sect.2. To put it briefly, Sect.3 g∈ex j f∈ex i
provides evidence for the existence of the mild so-
lution that has been established. Sect.4 provides The fundamental operations are defined as fol-
an example for helping in understanding. Finally, lows: The operation (ex i ⊕ ex j ) (t) is determined
Sect.5 reflects our conclusion. by:

2. Preliminaries (ex i ⊕ ex j ) (t) = sup min{ex i (t i ), ex j (t j )}.
t i +t j =t
Let H be a fuzzy number space and C(U, H) is a
space of all fuzzy numbers of all continuous func- The operation [βex] (t) is given by:
tions form ex : U → H with

 1
ex t , if β ̸= 0;
∥ex∥ C = sup ∥ex(t)∥ for ex ∈ C(U, H),  β
t∈U [βex] (t) = 1, if β = 0 and t = 0;

 0, if β = 0 and t ̸= 0.
1−µ
we denote, C τ,ω (U, H) = {ex| t ϱ e x(t) ∈
ϱ
C(U, H)}, with Here, β can be any real number.
ϱ 1−µ In this section, we introduce the Katugampola
t
= sup ∥ex(t)∥,
∥ex∥ C τ,ω fractional integral, defining them within the con-
t∈U ϱ
text of a finite positive interval [0, u] on the real
line, where 0 < u < ∞.
is
it is show that C(U, H) ⊂ C τ,ω (U, H), ∥ex∥ C τ,ω
a norm and The space C τ,ω (U, H) forms a set of To formulate the Katugampola fractional inte-
all fuzzy numbers as well. Consider E as a closed gral, we need to introduce several special func-
2
subspace within B, and L (U, E) be a space of all tions that play a crucial role in their definitions
and computations:
Lebesgue square integrable functions from U → E
Gamma function: The Gamma function Γ(u),
1
u 2
Z
2
2
∥e∥ 2 = ∥e(t)∥ dt for e ∈ L (U, E). extending the factorial to complex and real num-
L
0 bers, is defined by
2
It is clear that L (U, E) is also a Banach Space. Z ∞
e dt,
Γ(u) = t u−1 −t e x > 0, (2)
Definition 1. 23 A fuzzy number is a fuzzy set 0
e x : R → [0, 1] that satisfies the following condi- For positive integers m, it holds that
tions:
1. ex is normal, (i.e) there is a t 0 ∈ R such Γ(m) =(m − 1)!,
that ex(t 0 )= 1;
Γ(m) =mΓ(m − 1).
2. ex is a fuzzy convex in R, (i.e) for c ∈ [0, 1]
and for any t 1 , t 2 ∈ R Beta function: The Beta function B(u, v) is de-
e x(ct 1 + (1 − c)t 2 ) ≥ min{ex(t 1 ), ex(t 2 )}, fined as
3. ex is upper semi-continuous;
Z 1
4. The support of ex, which is the set of all u−1 v−1
B(u, v) = t (1 − t) dt, u, v > 0. (3)
t ∈ R such that ex(t) > 0, is bounded.
0
H represents the space of all fuzzy numbers on R. It can also be related to the Gamma function
84