Page 94 - IJOCTA-15-1

R. Hariharan, R. Udhayakumar / IJOCTA, Vol.15, No.1, pp.82-91 (2025)
t−ϵ ∞ t − s ϱ
ϱ
Z Z ω−1
+ω s ϱ−1 δM ω (δ)
Ξ ϵ,α e x(t) = e x 0 0 α ϱ
Γ(1 + ϱ(µ − 1) − ω) t − s ϱ ω
ϱ

× J δ F(s)dδds
t−ϵ ∞ t − s ϱ ϱ
ϱ
Z Z 1+ϱ(µ−1)−ω−1
× s ϱ+ω−2 2
0 α ϱ e x 0 ωL E t ϱ+ω+(ϱ −ϱ)(µ−1)
ϱ ω ≤
t ϱ 2+(ϱ−1)(µ−1)−ω
× ωδM ω (δ)J δ dδds
ϱ ω+ϱ−1
B t−ϵ ϱ , 1 + ϱ(µ − 1) − ω
t−ϵ ∞ t − s ϱ × t
ϱ
Z Z ω−1 ( ) ϱ
+ s ϱ−1 ωδM ω (δ) Γ(1 + ϱ(µ − 1) − ω)
0 α ϱ Z α
ϱ ϱ ω 1
t − s × + δM ω (δ)dδ
× J δ F(s)dδds Γ(1 + ω) 0
ϱ
ϱ ω L E ωt ϱ+τϱ(ω−1) ∥m(s)∥
ϵ e x 0 + B t−ϵ ϱ(1, ω)
=ωJ α ϱ 1+τ(ω−1) ( t )
ϱ Γ(1 + ϱ(µ − 1) − ω)
Z α
1
ϱ
t−ϵ ∞ t − s ϱ × + δM ω (δ)dδ
Z Z 1+ϱ(µ−1)−ω−1
× s ϱ+ω−2 δ Γ(1 + ω) 0
2
0 α ϱ ϱ+ω+(ϱ −ϱ)(µ−1)
ϱ ω ϱ ω e x 0 ωL E u
t ϵ ≤
× M ω (δ)J δ − α dδds ϱ 2+(ϱ−1)(µ−1)−ω
ϱ ϱ
ω+ϱ−1
t−ϵ ∞ t − s ϱ ( ) ϱ
ϱ
Z Z ω−1 B t−ϵ ϱ , 1 + ϱ(µ − 1) − ω
+ s ϱ−1 ωδM ω (δ) × t
0 α ϱ Γ(1 + ϱ(µ − 1) − ω)
ϱ ϱ ω ϱ ω ϱ+τϱ(ω−1)
t − s ϵ L E ωu ∥m(s)∥
× J δ − α F(s)dδds , + B t−ϵ ϱ(1, ω)
)
(
ϱ ϱ ϱ 1+τ(ω−1) t
Z α
1
+ δM ω (δ)dδ .
Γ(1 + ω)
ω ω
0
since J ϵ ϱ α is compact for ϵ ϱ α > 0,
ϱ ϱ From the above bounded condition, we can obtain
then the set Ξ ϵ,α (t) = {Ξ ϵ,α e x(t) : ex ∈ B r } is that the above right hand equation tends to be 0.
+
relatively compact in H. Furthermore, for any ∥Ξex(t) − Ξ ϵ,α e x(t)∥ C η,ex → 0 as α, ϵ → 0 , and we
e x ∈ B r , we obtain also obtain that R ∞ δM ω (δ)dδ = 1 from the
0 Γ(1+ω)
Lemma 2.
Thus, using the Arzela-Ascoli theorem, we
∥Ξex(t) − Ξ ϵ,α e x(t)∥ C η,ex demonstrate that Ξ(t) is relatively compact in H
for t ∈ U. Consequently, by the Schauder fixed
ϱ 1−µ
t
e x 0 ω point theorem (1), Ξ has a fixed point in B r ,
≤ sup
t∈U ϱ
Γ(1 + ϱ(µ − 1) − ω) which serves as the mild solution of fuzzy frac-
tional differential equations (1).
ϱ
t
Z Z ∞ t − s ϱ 1+ϱ(µ−1)−ω−1
× s ϱ+ω−2 δ
0 0 ϱ 4. Example
ϱ ω
t
× M ω (δ)J δ dδds Consider the following problem
ϱ
ϱ
t α t − s ϱ 2 1 2 −t
Z Z ω−1 
,
sin ex(t,ℓ)
∂
3 2
+ω s ϱ−1 δM ω (δ) ϱ D 0 + e x(t, ℓ) = ∂x 2 ex(t, ℓ) + e 1+e −t ,
0 0 ϱ 1 (7)
6
ϱ ϱ ω
ϱ I + ex(0, ℓ) = ex 0 (µ), x ∈ [0, h].
t − s
0
× J δ F(s)dδds
ϱ
2 1
,
Where ϱ D 3 2 is the Hilfer-Katugampola frac-
ϱ 1−µ
t
e x 0 ω 0 +
− sup
tional derivative of order τ = 2 3 , ω = 1 2 , t ∈
t∈U ϱ
Γ(1 + ϱ(µ − 1) − ω) (0, 1] = U 1 and ϱ > 0. The function ex(t)(ℓ) =
ϱ
Z t−ϵ Z ∞ t − s ϱ 1+ϱ(µ−1)−ω−1 e x(t, ℓ). F : U 1 × B → H is a fuzzy mapping and
× s ϱ+ω−2
0 α ϱ the continuous function F(t, ex(t)) is given by
ϱ ω
t −t
× δM ω (δ)J δ dδds e sin ex(t, ℓ)
ϱ F(t, ex(t))(ℓ) = −t ,
1 + e
88