Surveys in Mathematics and its Applications
ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 10 (2015), 149 – 157
FUZZY INITIAL VALUE PROBLEM
S. Melliani, L. S. Chadli, M. Atraoui and M. Bouaouid
Abstract. In this paper, we present the fuzzy solution concept of the initial value problem.
After introducing a metric between two fuzzy vectors, we prove that the system of fuzzy equations
has unique solution under some condition.
1
Introduction
The concept of fuzzy numbers and fuzzy arithmetic operations was first introduced
by Zadeh [11, 12], Dubois, and Prade [3]. We refer the reader to [4] for more
information on fuzzy numbers and fuzzy arithmetic. Fuzzy systems are used to
study a variety of problems including fuzzy metric spaces [8, 9], fuzzy differential
equations [6], fuzzy linear systems [10], and fuzzy partial differential equations [5].
In this paper, we give a concept solution for initial value problem of fuzzy linear
equations, we prove that the system of fuzzy equations has unique solution under
some condition.
2
Preliminaries
Throughout this paper, the notation Rn denotes n-dimensional Euclidean space and
the notation Rn×n denotes the set of all n × n real matrices. The norm in the space
Rn or Rn×n is regarded as ∥ • ∥∞ , that is
∥x∥∞ = max |xi |
1≤i≤n
for x = (x1 , . . . , xn )T ∈ Rn
and
∥A∥∞
⏐
⏐
⏐
⏐ n
⏐∑ ⏐
⏐
= max ⏐
aij ⏐⏐
1≤i≤n ⏐
⏐
j=1
for A = (aij ) ∈ Rn×n
2010 Mathematics Subject Classification: 03E72; 08A72; 34A07.
Keywords: Fuzzy number; Fuzzy initial value problem.
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Let a = (a1 , . . . , an )T and b = (b1 , . . . , bn )T . Define a ≤ b if and only if aj ≤ bj for
j = 1, 2, . . . , n. We denote
[a, b] = ([a1 , b1 ] , . . . , [an , bn ])T
and
In (R) =
{
if a ≤ b
[a, b] , a ≤ b, a ∈ Rn , b ∈ Rn
}
The addition, scale-multiplication and matrix-multiplication are defined as follows :
{
[ta, tb] t ≥ 0
[a, b] + [u, v] = [a + u, v + b] ,
[ta, tb] =
[tb,)ta] t < 0
(∑
∑n
n
T [a, b] =
j=1 t1j [aj , bj ] , . . . ,
j=1 tnj [aj , bj ]
where [a, b] ∈ In (R), [u, v] ∈ In (R), t ∈ R, T = (tij ) ∈ Rn×n . A metric d, in the
space In (R) is defined as
(
)
(
)
d [a, b] , [u, v] = max ∥a − u∥∞ , ∥b − v∥∞
for [a, b] ∈ In (R) and [u, v] ∈ In (R). Obviously, the above three operations within
In (R) are closed respectively, and In (R) is a complete metric space with the metric
d.
The α-cut of a fuzzy set u in Rn is denoted by [u]α i.e.
{
}
[u]α = t | t ∈ Rn , u(t) ≥ α ,
α>0
Let
{
}
Fn (R) = u | u is a fuzzy set in Rn , [u]α ∈ In (R) and [u]1 ̸= ∅
It is clear, F1 (R), denoted by F(R) in short, is the set of all closed and convex fuzzy
numbers. According to properties and representation theorems of closed and convex
fuzzy numbers, we can easily obtain the following two lemmas
(
)T
Lemma 1. Let u ∈ Fn (R). Then, u = (u1 , . . . , un )T and [u]α = [u1 ]α , [u2 ]α , . . . , [un ]α ,
α > 0, where uj ∈ F(R) (j = 1, 2, . . . , n).
Lemma 2. Let u, v ∈ Fn (R), α > 0, a ∈ R. Then [u + v]α = [u]α + [v]α , [au]α =
a [u]α .
Let Au =
n
(∑
j=1
a1j uj , . . . ,
n
∑
anj uj
)T
for A ∈ Rn×n , u = (u1 , u2 , . . . , un )T ∈ Fn (R)
j=1
where the addition and the multiplication are defined by Zadeh’s extension principle.
It is easy to see Au ∈ Fn (R). By Lemmas 1 and 2, we have
[Au]α = A [u]α
for u ∈ Fn (R) and A ∈ Rn×n
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Fuzzy initial value problem
Let
[u]α = [uα , uα ] =
([
]
[
])T
uα1 , uα1 , . . . , u]αn , uαn
where [u]α ∈ In (R), uα ∈ Rn , uα ∈ Rn , uα ≤ uα and uαj ∈ R, uαj ∈ R, uαj ≤ uαj .
Define a mapping ρ : Fn (R) × Fn (R) → R as follows :
(⏐
⏐ ⏐
⏐)
ρ(u, v) = sup d ([u]α , [v]α ) = sup max ⏐uαj − v αj ⏐ , ⏐uαj − v αj ⏐
α>0 1≤j≤n
α>0
We have
Lemma 3. Fn (R) is a complete metric space with the metric ρ.
Proof. According to Lemma 1, we obtain Fn (R) = F(R) × · · · × F(R). (F(R), d) is
a complete metric space with the metric d (see [...]), where
)
(
d(x, y) = sup |uα − v α | , |uα − v α |
α>0
for u ∈ F(R) and v ∈ F(R). Moreover,
(⏐
⏐)
⏐ ⏐
ρ(u, v) = sup max ⏐uαj − v αj ⏐ , ⏐uαj − v αj ⏐
α>0 j
(⏐
⏐)
⏐ ⏐
= max sup ⏐uαj − v αj ⏐ , ⏐uαj − v αj ⏐ = max d (uj , vj )
j
j
α>0
for u = (u1 , u2 , . . . , un )T ∈ Fn (R) and v = (v1 , v2 , . . . , vn )T ∈ Fn (R). Hence,
(Fn (R), ρ) is a complete metric space.
Lemma 4. ρ (Au, Av) ≤ ∥A∥∞ ρ (u, v) for u, v ∈ Fn (R), A ∈ Rn×n
Proof. According to the operation principle within In (R), Lemmas 1 and 2, for each
α > 0, we have
[Au]α = A [u]α = A [uα , uα ] = A ([uα1 , uα1 ] , . . . , [uαn , uαn ])T
(∑
)
n
n
∑
[ α α]
[ α α] T
=
a1j uj , uj , . . . ,
anj uj , uj
j=1
j=1
⎡
⎤
⎡
⎤
( ∑
)T
n
n
n
n
∑
∑
∑
∗
∗
= ⎣
a1j s1 (uj ),
a1j s1 (uj )⎦ , . . . , ⎣
anj sn (uj ),
anj sn (uj )⎦
j=1
where
j=1
{
uαj
si (uj ) =
uαj
j=1
j=1
aij ≥ 0
aij ≤ 0
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152
S. Melliani et al.
and
s∗i (uj ) = uαj + uαj − si (uj ) ,
j = 1, 2, . . . , n
i = 1, 2, . . . , n
Replacing u and uj (j = 1, 2, . . . , n) by v and vj (j = 1, 2, . . . , n) in the above
equalities, we can obtain a similar result. Therefore,
⏐
⏐
⏐
⏐
⏐∑
⏐
⏐∑
⏐)
(
n
n
⏐
⏐
⏐
⏐
α
α
∗
∗
⏐
⏐
⏐
d ([Au] , [Av] ) = max max ⏐
a1j (si (uj ) − si (vj ))⏐ , max ⏐
a1j (si (uj ) − si (vj ))⏐⏐
1≤i≤n ⏐
⏐ 1≤i≤n ⏐ j=1
⏐
j=1
⎛
⎞
n
(⏐
∑
⏐ ⏐
⏐)
⎝
= max
|aij |⎠ max ⏐uαj − v αj ⏐ , ⏐uαj − v αj ⏐
1≤i≤n
j=1
= ∥A∥∞ ρ (u, v)
which implies that
(
)
ρ (Au, Av) = sup d [Au]α , [Av]α ≤ ∥A∥∞ ρ (u, v)
α>0
The proof is completed.
Lemma 5. ρ (u + w, v + w) = ρ (u, v) for u, v, w ∈ Fn (R)
Proof.
ρ (u + w, v + w) = sup d ([u + w]α , [v + w]α )
α>0
= sup d ([u]α + [w]α , [v]α + [w]α )
α>0
= sup d ([uα + wα , uα + wα ] , [v α + wα , v α + wα ])
α>0
= sup max (∥uα − v α ∥∞ , ∥uα − v α ∥∞ )
α>0
= sup d ([u]α , [v]α ) = ρ (u, v)
α>0
In the following, we discuss the solution of fuzzy initial value problem
∂t u + Λ∂x u = Cu + D
u (x, 0) = u0 (x)
(2.1)
(
)
with Λ = diag λ1 , . . . , λn a square matrix for which its terms are fuzzy numbers,
C a real square matrix and D a fuzzy function.
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Fuzzy initial value problem
3
Fuzzy Initial Value Problem
We consider the initial value problem (2.1)
Definition 6. The system (2.1) is called hyperbolic in the sense that the eigenvalues
λi satify
λ1 ≻ λ2 ≻ · · · ≻ λn
i.e.
λi > λj
for all
i<j
with
[
]
λi = λi , λi
Along a characteristic Ci of the ith family
∂u ∑
∂u dx ∂u
∂u
du
=
+
=
+ λi
=
cik uk + di
dt
∂t
dt ∂x
∂t
∂x
k
The ith backward characteristic Ci through a point (X, T ) has an equation
x = X − λi (T − t) = γi (t, X, T )
(3.1)
obtained by solving the differential equation
dx
= λi
dt
with x = X for t = T . Then integrate along Ci
ui (X, T ) =
ui0 (X, T )
T
∫
+
hi (t, X, T ) dt
(3.2)
0
with
ui0 (X, T ) = u0 (γi (0, X, T ))
and
hi (t, X, T ) =
∑
cik uk + di
k
where in the integrand x has to be replaced by γi (t, X, T ). Formula (3.2) resembles
a system of integral equations except that the domain of integration is different for
each component of u. We write (3.2) symbolically as
u = W + Su
(3.3)
where W is the vector (considered known) with components
Wi (X, T ) = ui0 (X, T ) +
∫
T
di (γi (t, X, T ) , t) dt
0
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S. Melliani et al.
and S is the linear operator taking a vector u with components uk (x, t) into a vector
w = Su with component
∫ T∑
wi (X, T ) =
cik · uk (γi (t, X, T ) , t) dt
0
Lemma 7. For u, v ∈ Fn (R),
k
C ∈ Rn×n , we have
ρ (Cu, Cv) ≤ ∥C∥∞ ρ (u, v)
Proof. According to the operation principle within In (R), Lemmas 1 and 2, for each
α > 0, we have
[Cu]α = C [u]α = C [uα , uα ]
= C ([uα1 , uα1 ] , . . . , [uαn , uαn ])
⎛
⎞
n
n
∑
∑
]
]
[
[
= ⎝
c1j uαj , uαj , . . . ,
cnj uαj , uαj ⎠
j=1
j=1
T
= (C1 , . . . , Cn )
where
⎡
⎤
n
n
∑
∑
Ci = ⎣
cij s1 (uj ) ,
cij s∗1 (uj )⎦
j=1
(3.4)
j=1
{
uαj
si (uj ) =
uαj
cij ≤ 0
cij ≤ 0
and
s∗i (uj ) = uαj + uαj − si (uj )
for i, j = 1, . . . , n
Replacing u and uj (j = 1, 2, . . . , n)by v and vj in the above equalities, we can obtain
a similar result. Therefore for
⏐
⏐
⏐ n
⏐
∑
⏐
⏐
⏐
A = max ⏐
c1j (si (uj ) − si (vj ))⏐⏐
1≤i≤n ⏐
⏐
j=1
⏐
⏐
⏐
⏐∑
n
⏐
⏐
∗
∗
⏐
c1j (si (uj ) − si (vj ))⏐⏐
B = max ⏐
1≤i≤n ⏐
⏐
j=1
d ([Cu]α , [Cv]α ) = max (A, B)
≤ ∥C∥∞ max (|uαi − v αi | , |uαi − v αi |)
1≤i≤n
= ∥C∥∞ ρ (u, v)
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Fuzzy initial value problem
which implies that
ρ (Cu, Cv) = sup d ([Cu]α , [Cv]α ) ≤ ∥C∥∞ ρ (u, v)
α>0
The proof is completed.
Lemma 8. For u, v, z ∈ Fn (R), we have
ρ (u + z, v + z) = ρ (u, v)
Proof.
ρ (u + z, v + z) = sup d ([u + z]α , [v + z]α )
α>0
= sup d ([u]α + [z]α , [v]α + [z]α )
α>0
= sup max (∥uα − v α ∥∞ , ∥uα − v α ∥∞ )
α>0
= sup d ([u]α , [v]α ) = ρ (u, v)
α>0
Since, the norm ∥S∥ of the operator S obviously is bounded by the constant
q = T ∥C∥∞
In the following, the solution of a system of fuzzy linear equation u = W + Su. can
be regarded as the fixed point of a mapping :
u → gu = W + Su
Theorem 9. The mapping g has unique fixed point within Fn (R) if q < 1 (i.e. for
T sufficiently small)
Proof. By Lemmas 1 and 2, we know
ρ (gu, gv) ≤ qρ (u, v)
holds well for u, v ∈ Fn (R). Hence the mapping g is contractive with respect to the
metric ρ.
Lemma 3 shows that (Fn (R), ρ) is a complete metric space, therefore, there uniquely
exists a point u∗ ∈ Fn (R) such that gu∗ = W + Su∗ = u∗ , which completes the
proof.
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Surveys in Mathematics and its Applications 10 (2015), 149 – 157
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157
Fuzzy initial value problem
S. Melliani
L. S. Chadli
Department of Mathematics
Department of Mathematics
Sultan Moulay Slimane University,
Sultan Moulay Slimane University,
BP 523, 23000 Beni Mellal, Morocco
BP 523, 23000 Beni Mellal, Morocco
e-mail: melliani@fstbm.ac.ma
e-mail: chadli@fstbm.ac.ma
M. Atraoui
M. Bouaouid
Department of Mathematics
Department of Mathematics
Sultan Moulay Slimane University,
Sultan Moulay Slimane University,
BP 523, 23000 Beni Mellal, Morocco
BP 523, 23000 Beni Mellal, Morocco
e-mail: atraoui1@yahoo.fr
e-mail: mbouaouid@gmail.com
License
This work is licensed under a Creative Commons Attribution 4.0 International
License.
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