Page 74 - IJOCTA-15-4

B. Shiri et.al. / IJOCTA, Vol.15, No.4, pp.610-624 (2025)
and the fractional nabla difference is defined as Theorem 8. Suppose x i ∈ R F , i = 1, . . . , ν.
N
α
∇ x(t) = ∇ ∇ −(N−α) x(t), t ∈ N a+N , Then, H-difference
O −(1−⃗α) ⃗x(t−1) = ∇ −(1−⃗α) ⃗x(t−1)⊖T −(1−⃗α) ⃗x(t−1)
where ∇x(t) = x(t) − x(t − 1), and N a = {a, a +
1, · · · }. is feasible and
Definition 4. Let α ∈ (0, 1] and x : N a → R F . O −(1−⃗α) ⃗x(t − 1) =
Then, the fractional nabla H-difference is defined t−1
⃗ α X Γ(t − s − ⃗α) (32)
as ⃗x(s).
Γ(1 − ⃗α) Γ(t − s + 1)
α
∇ x(t) =∇∇ −(1−α) x(t) s=0
(25) Proof. First, we observe that for s = 0, · · · , t−1,
=∇ −(1−α) x(t) ⊖ ∇ −(1−α) x(t − 1),
(t − s − α i )Γ(t − s − α i ) Γ(t − s − α i )
for t ∈ N a+1 . If Equation (25) is feasible, we say < ,
(t − s)Γ(t − s) Γ(t − s)
it is fractional nabla H-differenceable.
and thus
It is useful to decompose ∇ −(1−α) x(t) into Γ(t + 1 − s − α i ) Γ(t − s − α i )
0 ≤ < .
peak and tail operators. Γ(t + 1 − s) Γ(t − s)
t
1 X Γ(t − s + (1 − α)) Therefore, the H-differences
−(1−α)
∇ x(t) = x(s)
Γ(1 − α) Γ(t − s + 1) 1 Γ(t − s − α i )
s=0 x i (s)⊖
Γ(1 − α i ) Γ(t − s)
t−1
1 X Γ(t − s + (1 − α))
=x(t) ⊕ x(s). 1 Γ(t + 1 − s − α i )x i (s)
Γ(1 − α) Γ(t − s + 1)
s=0 Γ(1 − α i ) Γ(t + 1 − s)
(26)
are feasible for s = 0, · · · , t − 1. Especially, their
Thus, x(t) is the peak operator and sum is also nabla H-differenceable and
t−1 O −(1−⃗α) x i (t − 1)
X Γ(t − s + (1 − α))
T −(1−α) x(t − 1) := x(s),
Γ(1 − α)Γ(t − s + 1) 1 X Γ(t − s − α i )
t−1
s=0 =
is the tail operator. We can rewrite Equation (26) Γ(1 − α i ) s=0 Γ(t − s)
as Γ(t + 1 − s − α i )
− x i (s)
∇ −(1−α) x(t) = x(t) ⊕ T −(1−α) x(t − 1). (27) Γ(t + 1 − s)
t−1
1 X Γ(t − s − α i ) (33)
=
3. Analysis of solution Γ(1 − α i ) Γ(t − s)
s=0
The first question is whether there exists an H- (t − s − α i )
differenceable solution for System Equation (1) if (1 − (t − s) ) x i (s)
x i (0) ∈ R F . We build a constructive method to
t−1
obtain the fuzzy solution of Equation (2) based = α i X Γ(t − s − α i ) x i (s).
on the H-difference operator. Equation (2) for Γ(1 − α i ) Γ(t − s + 1)
s=0
fuzzy inputs is
□
⃗ α
∇ ⃗x(t) = ⃗q(W⃗x(t − 1) ⃗ ⊕⃗p). (28)
Taking into account Theorem 8, System
Using Equation (25), we obtain Equation (31) can be written as
∇ −(1−⃗α) ⃗x(t) ⃗ ⊖∇ −(1−⃗α) ⃗x(t−1) = ⃗q(W⃗x(t−1) ⃗ ⊕⃗p). ⃗x(t) =O −(1−⃗α) ⃗x(t − 1) ⃗ ⊕⃗q(W⃗x(t − 1) ⃗ ⊕⃗p). (34)
(29)
Since the fuzzy difference is Hukuhara type, we Since Equation (34) expresses ⃗x(t) in terms of
⃗x(t − 1), ⃗x(t) can be obtained recursively.
can rewrite
∇ −(1−⃗α) ⃗x(t) = ∇ −(1−⃗α) ⃗x(t − 1) ⃗ ⊕⃗q(W⃗x(t − 1) ⃗ ⊕⃗p) Theorem 9. System Equation (28) subject to in-
(30) put ⃗x(0) = ⃗x 0 has a unique H-difference solution.
where ⃗ ⊕ and ⃗ ⊖ stand for the element-wise fuzzy Proof. We start with t = 1 to initiate the induc-
sum and H-difference, respectively. Using the de- tion process. Then,
composition in Equation (27), we get −(1−⃗α) −(1−⃗α)
⃗x(1) = ∇ ⃗x(0) ⃗ ⊖T ⃗x(0) ⃗ ⊕⃗q(W⃗x(0) ⃗ ⊕⃗p).
⃗x(t) ⃗ ⊕T −(1−⃗α) ⃗x(t − 1) = (35)
(31) From Theorem 8,
∇ −(1−⃗α) ⃗x(t − 1) ⃗ ⊕⃗q(W⃗x(t − 1) ⃗ ⊕⃗p).
∇ −(1−⃗α) ⃗x(0) ⃗ ⊖T −(1−⃗α) ⃗x(0) = ⃗α⃗x 0
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