Page 71 - IJOCTA-15-4

Analysis and analytical solution of incommensurate fuzzy fractional nabla difference systems...
(6) C µ∗ is a monotonically increasing func- Equivalently, the boundaries of an r-cut can
∗
tion and C is a monotonically decreasing define a fuzzy number, leading to the following
µ
function. equivalent definition.
∗
(7) C µ∗ and C are continuous functions.
µ
(8) C µ∗ and C µ ∗ are bounded and C µ∗ (r) ≤ Theorem 2. Let the functions C µ∗ : [0, 1] → R
∗
∗
µ
C (r) for all r ∈ [0, 1]. and C : [0, 1] → R satisfy properties 5–9 of The-
µ orem 1. Then, µ : R → [0, 1] defined by
∗
(9) C µ∗ (1) = C (1) = d.
µ
∗

sup{r : C (r) ≥ w}, w ≥ d,
µ
Proof. 1. Since µ is normal, there exists a unique µ(w) = (5)
sup{r : C µ∗ (r) ≤ w}, w ≤ d,
d ∈ R such that µ(d) = 1. For any r ∈ [0, 1], be-
cause 1 ≥ r, we have µ(d) ≥ r, so d ∈ C µ (r). is a fuzzy number.
Also, since there is at least one element (i. e., d) Proof. It suffices to prove that the function µ
in C µ (r), C µ (r) ̸= ∅. 2. By the definition of a satisfies the conditions of Definition Equation (1).
fuzzy number, there is a unique element d ∈ R The proof is rather straightforward. □
for which µ(d) = 1. So, when r = 1, C µ (1) =
{x ∈ R : µ(x) ≥ 1} = {d}. 3. Let r 1 ≤ r 2 . Due to the symmetry of the translated func-
∗
If x ∈ C µ (r 2 ), then µ(x) ≥ r 2 . Since r 2 ≥ r 1 , tions f 1 = d−C µ∗ and f 2 = C −d, we can simplify
µ
we have µ(x) ≥ r 1 , which implies x ∈ C µ (r 1 ). the fuzzy definition by defining it in a decomposed
Thus, C µ (r 2 ) ⊆ C µ (r 1 ). 4. Since µ is a convex form, similar to the approach in 21 and in parallel
function (by the definition of a fuzzy number), with the interval analysis. To do this, let d ∈ R.
for any r ∈ [0, 1], the set C µ (r) is convex. In R, Define the set U as:
convex sets are intervals. The support Supp(µ) is +
U = f : [0, 1] → R : f is m. d. c. on [0, 1],
compact. Since C µ (r) ⊆ Supp(µ) for all r ∈ [0, 1],
f(1) = 0} .
the closure C µ (r) is a closed and bounded inter-
val. The endpoints of this interval are precisely where “m. d. c.” abbreviates “monotonically de-
∗
C µ∗ (r) = inf C µ (r) and C (r) = sup C µ (r), so creasing and continuous”.
µ
∗
C µ (r) = [C µ∗ (r), C (r)]. 5. From 4, for each r ∈ Theorem 3. [21] Let C and C µ∗ be functions
∗
µ
µ
∗
[0, 1], the infimum C µ∗ (r) and supremum C (r) from [0, 1] to R. Then, f 1 , f 2 ∈ U if and only if
µ
are well-defined real numbers. So, C µ∗ : [0, 1] → the functions C = f 2 + d and C µ∗ = d − f 1 sat-
∗
∗
R and C : [0, 1] → R are well-defined functions. µ
µ isfy the properties specified in conditions 5–9 of
6. Let r 1 ≤ r 2 . From 3, C µ (r 2 ) ⊆ C µ (r 1 ).
Theorem 1.
Then, inf C µ (r 2 ) ≥ inf C µ (r 1 ) (because a subset
cannot have a smaller infimum than the super- Remark 3. Based on Theorem 3, a triple
set), so C µ∗ (r 2 ) ≥ C µ∗ (r 1 ), which means that C µ∗ (d, f 1 , f 2 ) represents a fuzzy number. To denote
increases monotonically. Similarly, sup C µ (r 2 ) ≤ the equivalence between this triple and a fuzzy
∗
∗
sup C µ (r 1 ), so C (r 2 ) ≤ C (r 1 ), which means that number µ, we use the symbol ∼, expressed as:
µ
µ
∗
C is monotonically decreasing. 7. Let r 0 ∈ [0, 1] µ ∼ (d, f 1 , f 2 ).
µ
and ϵ > 0. Since µ is upper continuous, for a small
δ > 0, if |r − r 0 | < δ, the change in the set C µ (r) Due to the assumption of compact support,
is small. Let r < r 0 . Then C µ (r) ⊇ C µ (r 0 ). As we can impose additional assumptions on the set
−
r → r , the infimum C µ∗ (r) approaches C µ∗ (r 0 ). U. To leverage the results in Ref. 23 for defining
0
+
Similarly, for r > r 0 , as r → r , C µ∗ (r) ap- arithmetic operations, we impose additional con-
0
proaches C µ∗ (r 0 ). The proof for the continuity ditions on the set U and define a new set S of
of C µ ∗ is similar, using the upper continuity of shape functions as follows:
+
µ and the properties of the supremum of the r- S = {f : [0, 1] → R : f is m. d. c. on [0, 1],
∗
cuts. 8. From 4, C µ (r) = [C µ∗ (r), C (r)] is a f(1) = 0, f(0) = 1 or f ≡ 0}, (6)
µ
bounded interval for all r ∈ [0, 1]. So, C µ∗ and
∗
C are bounded functions. Also, by the definition where “m. d. c.” abbreviates “monotonically de-
µ
∗
of infimum and supremum, C µ∗ (r) ≤ C (r) for creasing and continuous”.
µ
all r ∈ [0, 1]. 9. From 2, C µ (1) = {d}. Then, Theorem 4. Suppose Supp(µ) = [a, b]. For each
∗
inf C µ (1) = C µ∗ (1) = d and sup C µ (1) = C (1) = µ ∼ (d, f 1 , f 2 ), there exist f 1 , f 2 ∈ S such that
˜ ˜
µ
d. □ ˜ ˜
f 1 = (d − a)f 1 and f 2 = (b − d)f 2 .
Remark 2. In comparison, previous definitions
˜
Proof. If d − a ̸= 0, define f 1 = f 1 ; otherwise,
do not satisfy properties 2 and 9. This paper d−a
˜
˜
f 2
presents a minor modification that addresses this set f 1 = 0. If b − d ̸= 0, define f 2 = b−d ; other-
˜
shortcoming. wise, set f 2 = 0. □
613