Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 184950, 9 pages
http://dx.doi.org/10.1155/2013/184950
Research Article
About Projections of Solutions for Fuzzy Differential Equations
Moiseis S. Cecconello,1 Jefferson Leite,2 Rodney C. Bassanezi,3 and Joao de Deus M. Silva4
1
DMAT-ICET-UFMT, 78075-202 Cuiabá, MT, Brazil
DEMAT-CCN-UFPI, 64063040 Teresina, PI, Brazil
3
CMCC-UFABC, 09210-170 Santo André, SP, Brazil
4
CCET-UFMA, 65085-558 São Luiı́s, MA, Brazil
2
Correspondence should be addressed to Jefferson Leite; jleite@ufpi.edu.br
Received 13 February 2013; Accepted 23 April 2013
Academic Editor: Ch. Tsitouras
Copyright © 2013 Moiseis S. Cecconello et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
In this paper we propose the concept of fuzzy projections on subspaces of F(R𝑛 ), obtained from Zadeh’s extension of canonical
projections in R𝑛 , and we study some of the main properties of such projections. Furthermore, we will review some properties of
fuzzy projection solution of fuzzy differential equations. As we will see, the concept of fuzzy projection can be interesting for the
graphical representation of fuzzy solutions.
1. Introduction
Consider the set 𝑈 ⊂ R𝑛 . Denote by F(𝑈) the set formed by
the fuzzy subsets of 𝑈 whose subsets have support compacts
in 𝑈. Some properties for metrics F(𝑈) can be found in [1].
If 𝐴 is a subset of 𝑈, we will use the notation 𝜒𝐴 to indicate
a membership function for the fuzzy set called membership
function or crisp of 𝑈.
Consider the autonomous equation defined by
𝑑𝑥
= 𝑓 (𝑥) ,
𝑑𝑡
(1)
where 𝑓 : 𝑈 ⊂ R𝑛 → R𝑛 is a sufficiently smooth function.
For each 𝑥𝑜 ∈ 𝑈, denote by 𝜑𝑡 (𝑥𝑜 ) the deterministic solution
(1) with initial condition 𝑥𝑜 . Here we are assuming that the
solution is defined for all 𝑡 ∈ R+ . The function 𝜑𝑡 : 𝑈 → 𝑈
will be called deterministic flow.
To consider initial conditions with inaccuracies modeled
by fuzzy sets [2], consider the proposed Zadeh’s extension
̂ 𝑡 : F(𝑈) → F(𝑈), which takes the
𝜑𝑡 , the application 𝜑
̂ 𝑡 (x𝑜 ). In the context
fuzzy set x𝑜 ∈ F(𝑈) and the fuzzy set 𝜑
̂ 𝑡 of fuzzy flow. Given
of this paper we call the application 𝜑
̂ 𝑡 (x𝑜 ) is a fuzzy solution to (1) whose initial
x𝑜 ∈ F(𝑈), we say 𝜑
condition is the fuzzy set x𝑜 .
The conditions for existence of fuzzy equilibrium points
and the nature of the stability of such spots were first
presented in [2]. The concepts of stability and asymptotic
stability for fuzzy equilibrium points are similar to those of
equilibrium points of deterministic solutions, and stability
conditions for fuzzy equilibrium points can be found in [2].
Conditions for the existence of periodic fuzzy solutions and
the stability of such solutions can be found in [3].
In this paper, we propose the concept of fuzzy projections
on subspaces of F(R𝑛 ), obtained from Zadeh’s extension
defined canonical projections in R𝑛 , and study some of
the main properties of such projections. Furthermore, we
review some properties of fuzzy projection solution of fuzzy
differential equations. As we will see, the concept of fuzzy
projection can be interesting for the graphical representation
of fuzzy solutions.
2. Projections in Fuzzy Metric Spaces
We restrict our analysis to the set F(𝑋) whose elements are
subsets of a fuzzy set 𝑋 whose 𝛼-levels are compact and
nonempty subsets in 𝑋. The fuzzy subsets that are F(𝑋)
will be denoted by bold lowercase letters to differentiate the
elements 𝑋. So x ∈ F(𝑋) if and only if [x]𝛼 is compact and
nonempty subset for all 𝛼 ∈ [0, 1].
We can define a structure of metric spaces in F(𝑋) by the
Hausdorff metric for compact subsets of 𝑋. Let K(𝑋) be the
set formed by nonempty compact subsets of the metric space
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Journal of Applied Mathematics
(𝑋, 𝑑). Given two sets 𝐴, 𝐵 in K(𝑋), the distance between
them can be defined by
F(R𝑛+𝑚 ) in F(R𝑛 ), as it can be identified with the subset
F(R𝑛 ) × 𝜒{0} . Similarly the projection 𝑃𝑛 satisfies:
dist (𝐴, 𝐵) = sup inf 𝑑 (𝑎, 𝑏) .
𝑃̂𝑛 (𝑃̂𝑛 (z)) = 𝑃̂𝑛 (z) .
(2)
𝑎∈𝐴 𝑏∈𝐵
The distance between sets defined above is a pseudometric
to K(𝑋) since dist (𝐴, 𝐵) = 0 if and only if 𝐴 ⊆ 𝐵, not
necessarily equal value. However, Hausdorff distance between
𝐴, 𝐵 ∈ K(𝑋) defined by
𝑑𝐻 (𝐴, 𝐵) = max {sup inf 𝑑 (𝑎, 𝑏) , sup inf 𝑑 (𝑎, 𝑏)}
𝑏∈𝐵 𝑎∈𝐴
𝑎∈𝐴 𝑏∈𝐵
(3)
= max {dist (𝐴, 𝐵) , dist (𝐵, 𝐴)}
is a metric for all K(𝑋). so that (K(𝑋), 𝑑𝐻) is a metric space.
It is also worth that (𝑋, 𝑑) is a complete metric space, so
(K(𝑋), 𝑑𝐻) is also a complete metric space [4].
Through the Hausdorff metric 𝑑𝐻, we can define a metric
for all F(𝑋). Here we denote it by 𝑑∞ . Given two points u, k ∈
F(𝑋), the distance between u, k is defined by
𝑑∞ (u, k) = sup 𝑑𝐻 ([u]𝛼 , [k]𝛼 ) .
(4)
𝛼∈[0,1]
It is not difficult to show that the distance defined above
satisfies the properties of a metric and thus (F(𝑋), 𝑑∞ ) is a
metric space.
Nguyen’s theorem provides an important link between 𝛼levels image of fuzzy subsets and the image of his 𝛼-levels by
a function 𝑓 : 𝑋 × 𝑌 → 𝑍. According to [5], if 𝑋 ⊆ R𝑛
and 𝑌 ⊆ R𝑚 and 𝑓 : 𝑋 → 𝑌 is continuous, then Zadeh’s
extension 𝑓̂ : F(𝑋) → F(𝑌) is well defined and is worth
𝛼
[𝑓̂ (u)] = 𝑓 ([u]𝛼 )
(5)
𝑛+𝑚
2.1. Projections Fuzzy. Consider the application 𝑃𝑛 : R
→
R𝑛 that for each (𝑥, 𝑦) ∈ R𝑛+𝑚 associates point 𝑃𝑛 (𝑥, 𝑦) = 𝑥 ∈
R𝑛 .
Provided that R𝑛 can be characterized as a subset of R𝑛+𝑚
by identifying it with the subset R𝑛 × {0}, then the application
𝑃𝑛 can be seen as the projection of R𝑛+𝑚 on the set R𝑛 . For
this reason, we say that 𝑥 is the projection in R𝑛 ; the point
(𝑥, 𝑦) ∈ R𝑛+𝑚 .
Notice that a point (𝑢, V) is in the image of 𝑃𝑛 if and
only if V = 0. Furthermore, 𝑃𝑛 (𝑥, 𝑦) = 𝑥 for all 𝑦 ∈ R𝑚 .
Thus, given a point z ∈ F(R𝑛+𝑚 ), with membership function
𝜇z : R𝑛+𝑚 → [0, 1], the image 𝑃̂𝑛 (z), obtained by Zadeh’s
extension projection 𝑃𝑛 , has the membership function
𝜇𝑃̂𝑛 (z) (𝑥) = sup 𝜇z (𝑥, V) .
V∈R𝑚
Based on this, we can define the projection of fuzzy
z ∈ F(R𝑛+𝑚 ) in F(R𝑛 ) as the point x ∈ F(R𝑛 ) with a
membership function
𝜇x (𝑥) = sup 𝜇z (𝑥, 𝑎) .
𝑎∈R𝑚
𝜇y (𝑦) = sup 𝜇z (𝑎, 𝑦)
𝑎∈R𝑛
(9)
which we call fuzzy projection z in F(R𝑚 ). Thus the application 𝑃̂𝑚 : F(R𝑛+𝑚 ) → F(R𝑚 ) can be viewed as a fuzzy
projection F(R𝑛+𝑚 ) in F(R𝑚 ).
Here are some examples.
Example 1. Let a ∈ F(R𝑛 ) and b ∈ F(R𝑚 ). We can define
z = (a, b) ∈ F(R𝑛+𝑚 ) with membership function
𝜇z (𝑥, 𝑦) = min {𝜇a (𝑥) , 𝜇b (𝑦)} .
(10)
The image of z by applying 𝑃̂𝑛 , in this case, has a
membership function:
𝜇𝑃̂𝑛 (z) (𝑥) = sup min {𝜇a (𝑥) , 𝜇b (V)} .
V∈R𝑚
(11)
Since min {𝜇a (𝑥), 𝜇b (V)} ≤ 𝜇a (𝑥), so,
sup min {𝜇a (𝑥) , 𝜇b (V)} ≤ 𝜇a (𝑥) .
(12)
As b ∈ F(R𝑚 ), so V ∈ R𝑚 so that 𝜇b (V) = 1. So, the fuzzy
projection x of z about F(R𝑛 ) has a membership function:
𝜇x (𝑥) = sup min {𝜇a (𝑥) , 𝜇b (V)} = 𝜇a (𝑥) .
V∈R𝑚
(13)
In Figure 1, the membership functions of z ∈ F(R2 ),
defined from a and b ∈ F(R) and your fuzzy projection in
F(R), respectively, can be seen. In this figure,
𝜇a (𝑥) = 𝜇b (𝑥) = max {1 − 𝑥2 , 0} .
(14)
With similar argument, we can show that b ∈ F(R𝑚 ) is a
fuzzy projection of z in F(R𝑚 ).
We can also define x = (a, b) ∈ F(R𝑛+𝑚 ) through the
𝑡-𝑛𝑜𝑟𝑚 product, that is,
𝜇z (𝑥, 𝑦) = 𝜇a (𝑥) 𝜇b (𝑦) .
(6)
The application 𝑃̂𝑛 : F(R𝑛+𝑚 ) → F(R𝑛 ), obtained by
Zadeh’s extension of 𝑃𝑛 , that for each z ∈ F(R𝑛+𝑚 ) associates
the point 𝑃̂𝑛 (z) ∈ F(R𝑛 ) can be seen as a projection of
(8)
We also consider the function 𝑃𝑚 : R𝑛+𝑚 → R𝑚 that for
all (𝑥, 𝑦) ∈ R𝑛+𝑚 associates the point 𝑃𝑛 (𝑥, 𝑦) = 𝑦 ∈ R𝑚 .
In this case, the image of a point z ∈ F(R𝑛+𝑚 ), with the
membership function 𝜇z : R𝑛+𝑚 → [0, 1], is a point y ∈
F(R𝑚 ) with the membership function
V∈R𝑚
for all 𝛼 ∈ [0, 1] and u ∈ F(𝑋).
(7)
(15)
The projection of z in F(R𝑛 ) has a membership function:
sup 𝜇x (𝑥, V) = sup 𝜇a (𝑥) 𝜇b (V) = 𝜇a (𝑥) .
V∈R𝑚
V∈R𝑚
(16)
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3
Moreover, the projection F(R𝑚 ) has a membership function:
𝑢∈R𝑛
Similarly, we can show that fuzzy projections z = (a, b) in
F(R𝑛 ) and F(R𝑚 ) for all 𝑡-𝑛𝑜𝑟𝑚 Δ are, respectively, a and b.
First, for any 𝑡-𝑛𝑜𝑟𝑚 Δ, we have
Δ (𝜇a (𝑥) , 𝜇b (𝑦)) ≤ Δ (𝜇a (𝑥) , 1) = 𝜇a (𝑥) .
1
(17)
𝜇x
sup 𝜇x (𝑢, 𝑦) = sup 𝜇a (𝑢) 𝜇b (𝑦) = 𝜇b (𝑦) .
𝑢∈R𝑛
0.5
0
(18)
1
So,
sup Δ (𝜇a (𝑥) , 𝜇b (V)) ≤ 𝜇a (𝑥) .
V∈R𝑚
y
0
−1
(19)
V∈R𝑚
−1
1
But the ultimate is reached if we take V ∈ R𝑛 so that 𝜇b (V) = 1.
Then, the projection of z = (a, b) in F(R𝑛 ) has membership
function
𝜇x (𝑥) = sup Δ (𝜇a (𝑥) , 𝜇b (V)) = 𝜇a (𝑥) ,
0
1
x
0.8
0.6
𝜇
(20)
0.4
for all 𝑡-𝑛𝑜𝑟𝑚 Δ.
0.2
2
Example 2. Consider z ∈ F(R ) determined by membership
function
2
2
𝜇z (𝑥, 𝑦) = max {1 − 𝑥 − 2𝑦 , 0} .
𝜇x (𝑥) = sup 𝜇z (𝑥, V) = max {1 − 𝑥2 , 0} ,
𝜇y (𝑦) = sup 𝜇z (𝑢, 𝑦) = max {1 − 2𝑦2 , 0} .
−1
0.5
−0.5
1
1.5
x
(21)
For this case, we have the fuzzy projections x and y on
F(R), respectively, determined by
V∈R𝑚
−1.5
Figure 1: Membership function of z and a respectively.
We can prove that dist ([y]𝛼 , [x]𝛼 ) ≥ dist ([y]𝛼 , [x]𝛼 ). Therefor,
(22)
𝑑∞ (x, y) ≤ 𝑑∞ (x, y) .
(24)
𝑢∈R𝑛
In Figure 2 we can see the membership functions z and x,
respectively.
Proposition 3. Let x = 𝑃̂𝑛 (x) and y = 𝑃̂𝑛 (y), with x and y ∈
F(R𝑛+𝑚 ). The distance between the fuzzy projections x and y
is always limited by the distance between x and y.
Proof. In fact, for all 𝛼 ∈ [0, 1] we have
The fuzzy projection p ∈ F(R𝑛 ) to a point p ∈
F(R𝑛+𝑚 ) satisfies another important property of the projections. Namely, the projection p is the point that minimizes the
distance between the point p ∈ F(R𝑛+𝑚 ) and the set F(R𝑛 ),
the latter set is considered as a subset of F(R𝑛+𝑚 ).
Proposition 4. The fuzzy projection p in F(R𝑛 ) of p ∈
F(R𝑛+𝑚 ) is such that
𝑑∞ (p, p) = inf 𝑛 𝑑∞ (p, z) .
𝛼
dist ([x]𝛼 , [y] )
z∈F(R )
Proof. First, let us note the abuse of notation in the statement.
The term 𝑑∞ (p, z) only makes sense because we can see
F(R𝑛 ) as a subset of F(R𝑛+𝑚 ). Provided that [p]𝛼 ⊂ R𝑛+𝑚
and [p]𝛼 ⊂ R𝑛 , for 𝑥 ∈ R𝑛 and 𝑦 = (𝑦1 , 𝑦2 ) ∈ R𝑛+𝑚 , we have
= sup inf 𝛼 ‖𝑎 − 𝑏‖
𝑎∈[x]𝛼 𝑏∈[y]
=
sup
(𝑎1 ,𝑎2 )∈[x]
𝛼
󵄩
󵄩2 󵄩
󵄩2
inf 𝛼 √󵄩󵄩󵄩𝑎1 − 𝑏1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑎2 − 𝑏2 󵄩󵄩󵄩
(𝑏1 ,𝑏2 )∈[y]
󵄩
󵄩2
≥ sup
inf √󵄩󵄩󵄩𝑎1 − 𝑏1 󵄩󵄩󵄩
𝛼 (𝑏 ,𝑏 )∈ y 𝛼
(𝑎1 ,𝑎2 )∈[x] 1 1 [ ]
󵄩
󵄩2
= sup inf 𝛼 √󵄩󵄩󵄩𝑎1 − 𝑏1 󵄩󵄩󵄩
𝛼 𝑏 ∈ y
𝑎1 ∈[x] 1 [ ]
𝛼
= dist ([x]𝛼 , [y] ) .
(25)
󵄩
󵄩
󵄩2 󵄩 󵄩2
󵄩󵄩
󵄩󵄩𝑥 − 𝑦󵄩󵄩󵄩 = √󵄩󵄩󵄩𝑥 − 𝑦1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
(23)
(26)
since
󵄩
󵄩2 󵄩 󵄩2
dist ([p]𝛼 , [p]𝛼 ) = sup inf 𝛼 √󵄩󵄩󵄩𝑦1 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
𝛼 𝑥∈ p
[]
𝑦∈[p]
󵄩 󵄩
= sup 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 .
𝛼
𝑦∈[p]
(27)
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Journal of Applied Mathematics
for all 𝑥 ∈ [q]𝛼 . The second property follows directly from
the projection inequality
𝜇x
1
󵄩
󵄩2 󵄩 󵄩2 󵄩 󵄩
󵄩
󵄩󵄩
󵄩󵄩𝑧1 − 𝑦󵄩󵄩󵄩 = √󵄩󵄩󵄩𝑧1 − 𝑦1 󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 > 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 ,
for all 𝑦 = (𝑦1 , 𝑦2 ) ∈ [p]𝛼 . Thus in both cases we have to
0.5
0
1
y
0
1
0
−1
x
−1
1
0.6
We can also define fuzzy projections z ∈ F(𝑈 × 𝑃) in
F(𝑈) and F(𝑃), where 𝑈 ⊂ R𝑛 and 𝑃 ⊂ R𝑚 . In this case,
the supremum in membership functions (8) and (9) is taken
on the sets 𝑃 and 𝑈, respectively, and properties shown above
metrics remain valid.
We can also consider the projection 𝜋𝑖 : R𝑛 → R from a
point 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) ∈ R𝑛 in 𝑖th coordinate axis; that
is, 𝜋𝑖 (𝑥) = 𝑥𝑖 . As shown before, the projection of Zadeh’s
̂ 𝑖 : F(R𝑛 ) → F(R)
extension 𝜋𝑖 defines the application 𝜋
that we call for the 𝑖th fuzzy projection of F(R𝑛 ) on F(R).
Thus, given a point x ∈ F(R), the 𝑖th fuzzy projection of x
on F(R) is a point x𝑖 with membership function given by
0.4
0.2
0.5
−0.5
1
1.5
x
Figure 2: Membership function of z and x, respectively.
Moreover, we have
󵄩
󵄩2 󵄩 󵄩2
dist ([p]𝛼 , [p]𝛼 ) = sup inf 𝛼 √󵄩󵄩󵄩𝑦1 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 . (28)
𝛼 𝑦∈ p
[]
𝑥∈[p]
𝛼
(33)
Therefore, we have 𝑑𝐻([p]𝛼 , [q]𝛼 ) ≥ 𝑑𝐻([p]𝛼 , [p]𝛼 ). Thus,
we can conclude that, for all q ∈ F(R𝑛 ), 𝑑∞ (p, q) ≥
𝑑∞ (p, p), which proves the assertion.
𝜇
−1
󵄩
󵄩2 󵄩 󵄩2
dist ([p]𝛼 , [q]𝛼 ) = sup inf 𝛼 √󵄩󵄩󵄩𝑦1 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
𝛼 𝑥∈ p
[]
𝑦∈[p]
󵄩 󵄩
≥ sup 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
𝛼
𝑦∈[p]
= dist ([p]𝛼 , [p]𝛼 ) .
0.8
−1.5
(32)
𝛼
𝑚
Now, since 𝑥 ∈ [p] , so, (𝑥, 𝑧) ∈ [p] for some 𝑧 ∈ R , where
we have the inequality
󵄩
󵄩2 󵄩 󵄩2
dist ([p]𝛼 , [p]𝛼 ) = sup inf 𝛼 √󵄩󵄩󵄩𝑦1 − 𝑥󵄩󵄩󵄩 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
𝛼 𝑦∈ p
[]
𝑥∈[p]
󵄩 󵄩
≤ ‖𝑧‖ ≤ sup 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 .
𝛼
𝑦∈[p]
𝑥∈R𝑛
𝑥𝑖 =𝑎
(29)
𝑑𝐻 ([p]𝛼 , [p]𝛼 ) = max {dist ([p]𝛼 , [p]𝛼 ) , dist ([p]𝛼 , [p]𝛼 )}
= dist ([p]𝛼 , [p]𝛼 ) .
(30)
Let q ∈ F(R𝑛 ) such that q ≠ p. This implies that
[q] ≠ [p]𝛼 , for some 𝛼 ∈ [0, 1]. Consequently, there 𝑦 =
(𝑦1 , 𝑦2 ) ∈ [p]𝛼 such that 𝑦1 ∉ [q]𝛼 or exists 𝑧1 ∈ [q]𝛼 such
that 𝑧 = (𝑧1 , 𝑧2 ) ∉ [p]𝛼 , for all 𝑧2 ∈ R𝑚 . Namely, 𝑧1 ∉ [p]𝛼 .
For the first case, we have
𝛼
(31)
(34)
Again, if x = (a1 , a2 , . . . , a𝑛 ) is defined by fuzzy Cartesian
product, then 𝑖th fuzzy projection of x ∈ F(R𝑛 ) in F(R) is a
point a𝑖 . For simplicity, consider x ∈ R3 defined by
𝜇x (𝑥, 𝑦, 𝑧) = Δ (Δ (𝜇a1 (𝑥) , 𝜇a2 (𝑦)) , 𝜇a3 (𝑧)) .
Thus, the Hausdorff distance between [p]𝛼 and [p]𝛼 in
this case is
√󵄩󵄩󵄩𝑥 − 𝑦1 󵄩󵄩󵄩 2 + 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩 2 > 󵄩󵄩󵄩𝑦2 󵄩󵄩󵄩
󵄩
󵄩
󵄩 󵄩
󵄩 󵄩
𝜇x𝑖 (𝑎) = sup 𝜇x (𝑥) .
(35)
By the properties of 𝑡-𝑛𝑜𝑟𝑚, it follows that
Δ (Δ (𝜇a1 (𝑥) , 𝜇a2 (𝑦)) , 𝜇a3 (𝑧))
≤ Δ (Δ (𝜇a1 (𝑥) , 𝜇a2 (𝑦)) , 1)
(36)
= Δ (𝜇a1 (𝑥) , 𝜇a2 (𝑦))
≤ 𝜇a2 (𝑦) ,
for all 𝑥, 𝑦, 𝑧 ∈ R.
Thus, the second fuzzy projection x on F(R) is the point
x2 where the membership function is
𝜇x2 (𝑎) = sup 𝜇x (𝑥) .
𝑥∈R3
𝑥2 =𝑎
(37)
For the previous inequality, we have
𝜇x2 (𝑎) = sup 𝜇x (𝑥) ≤ 𝜇a2 (𝑎) .
𝑥∈R3
𝑥2 =𝑎
(38)
Journal of Applied Mathematics
5
Taking 𝑥 and 𝑧 such that 𝜇a1 (𝑥) = 𝜇a3 (𝑧) = 1, equality is
attained in the supremum, and hence,
𝜇x2 (𝑎) = sup 𝜇x (𝑥) = 𝜇a2 (𝑎) .
(39)
𝑥∈R3
𝑥2 =𝑎
Induction proves the general case in which x ∈ F(R𝑛 ).
Through expression (8), we can determine the 𝛼-levels of
fuzzy projection x ∈ F(R𝑛 ) to a point z ∈ F(R𝑛+𝑚 ). Indeed,
if 𝜇x (𝑥) ≥ 𝛼, so, 𝑦 ∈ R𝑚 such that 𝜇z (𝑥, 𝑦) ≥ 𝛼 so that (𝑥, 𝑦) ∈
[z]𝛼 . The reciprocal is also true, because if 𝜇z (𝑥, 𝑦) ≥ 𝛼, then
by (8), 𝜇x (𝑥) ≥ 𝛼. Thus, we conclude that:
𝛼
𝛼
𝑥 ∈ [x] ⇐⇒ (𝑥, 𝑦) ∈ [z]
𝑚
for some 𝑦 ∈ R ,
(40)
or
[x]𝛼 = {𝑥 ∈ R𝑛 : (𝑥, 𝑦) ∈ [z]𝛼 } .
(41)
Since applying 𝜋𝑖 is continuous, we can use the equality
(5) to show that the 𝑖th fuzzy projection x𝑖 ∈ F(R) of x ∈
F(R𝑛 ) has 𝛼-levels:
𝛼
[x𝑖 ] = {𝑎 ∈ R : 𝑥 ∈ [x]𝛼 , 𝑥𝑖 = 𝑎} .
(42)
3. Projection of Fuzzy Solutions
3.1. Projection on the Coordinate Axes. Consider the flow 𝜑𝑡 :
𝑈 ⊂ R𝑛 → 𝑈 generated by the autonomous equation
𝑑𝑥
= 𝑓 (𝑥) ,
𝑑𝑡
(43)
where 𝜑(𝑖)
𝑡 : 𝑈 → R is the projection of the deterministic
flow 𝑖th coordinate axis; that is, 𝜑(𝑖)
𝑡 (𝑥𝑜 ) is the 𝑖th solution
component 𝜑𝑡 (𝑥𝑜 ), or even 𝜑(𝑖)
(𝑥
𝑜 ) is the solution of the
𝑡
equation
𝑑𝑥𝑖
= 𝑓𝑖 (𝑥) ,
𝑑𝑡
𝑥 (0) = 𝑥𝑜 .
(44)
By applying Zadeh’s extension to 𝜑(𝑖)
𝑡 , we have the applicâ (𝑖)
:
F(𝑈)
→
F(R)
that
for
each
x𝑜 ∈ F(𝑈) associates
tion 𝜑
𝑡
̂ (𝑖)
(x
)
∈
F(R).
As
in
the
deterministic
case, we
the image 𝜑
𝑜
𝑡
̂ 𝑡 (𝑖) : F(𝑈) → F(R) ia an 𝑖th
show that the application 𝜑
̂ 𝑡 : F(𝑈) → F(𝑈) on F(R).
fuzzy projection to fuzzy flow 𝜑
̂ (𝑖)
Proposition 5. The application 𝜑
𝑡 : F(𝑈) → F(R) is 𝑖th
̂ 𝑡 : F(𝑈) → F(𝑈) on F(R).
fuzzy projection of fuzzy flow 𝜑
Proof. Let x𝑜 ∈ F(𝑈). By definition, 𝑖th fuzzy projection
̂ 𝑖 (̂
̂ 𝑡 (x𝑜 ) on F(R) is the point x𝑖 = 𝜋
𝜑𝑡 (x𝑜 )). Since the
𝜑
projection is a continuous map, then it is worth
𝛼
𝛼
𝛼
= {𝜑(𝑖)
𝑡 (𝑥𝑜 ) : 𝑥𝑜 ∈ [x𝑜 ] }
(45)
𝛼
𝛼
(46)
where x𝑖 is 𝑖th fuzzy projection of the fuzzy equilibrium point
x𝑒 .
Consider just a few examples of the results presented
previously.
Example 6. The autonomous equation
𝑑𝑥1
= 𝑥2 ,
𝑑𝑡
𝑥1 (0) = 𝑥01 ,
𝑑𝑥2
= −𝑥1 ,
𝑑𝑡
𝑥2 (0) = 𝑥02 ,
(47)
determines the flow two-dimensional 𝜑𝑡 : R2 → R2 , 𝜑𝑡 =
(2)
(𝜑(1)
𝑡 , 𝜑𝑡 ), given by
𝜑(1)
𝑡 (𝑥01 , 𝑥02 ) = 𝑥01 cos 𝑡 + 𝑥02 sen 𝑡,
𝜑(2)
𝑡 (𝑥01 , 𝑥02 ) = 𝑥02 cos 𝑡 − 𝑥01 sen 𝑡.
(48)
We have already shown in [3] that the fuzzy solution
̂ 𝑡 (x𝑜 ) this equation is periodic for any choice of initial con𝜑
dition x𝑜 ∈ F(R2 ). According to the previous proposition,
̂ 𝑡 : F(R2 ) → F(R2 ) on F(R) are
projections of fuzzy 𝜑
obtained by taking extensions of components Zadeh 𝜑(1)
𝑡 and
𝜑(2)
.
𝑡
Figure 3 shows the time evolution of the fuzzy projection
̂ 𝑡 (x𝑜 ) on 𝑥 and 𝑦, respectively. Take the initial condition
of 𝜑
𝑥0 defined by the membership function.
𝜇x𝑜 (𝑥, 𝑦) = max {1 − (𝑥 − 3)2 − 𝑦2 , 0} .
(49)
Example 7. Consider the epidemiological model 𝑆𝐼 defined
by equations
𝑑𝑆
= −𝑟𝑆𝐼,
𝑑𝑡
𝑑𝐼
= 𝑟𝑆𝐼,
𝑑𝑡
𝑆 (0) = 𝑆𝑜 > 0,
𝐼 (0) = 𝐼𝑜 > 0,
(50)
𝑆 = 𝐼 + 𝑁.
= 𝜑(𝑖)
𝑡 ([x𝑜 ] ) .
is proved.
𝜇x𝑖 (𝑥) = 𝜇𝑥̂𝑒(𝑖) (x𝑜 ) (𝑥) ,
𝛼
[x𝑖 ] = 𝜋𝑖 (𝜑𝑡 ([x𝑜 ] )) = {𝜋𝑖 (𝜑𝑡 (𝑥𝑜 )) : 𝑥𝑜 ∈ [x𝑜 ] }
𝛼
Then, [̂
𝜑(𝑖)
𝑡 (x𝑜 )]
We showed in [3] that the equilibrium point 𝑥𝑒 deterministic flow 𝜑𝑡 : 𝑈 → 𝑈 depends on the initial condition
̂𝑡 :
𝑥𝑜 ∈ 𝑈; then the equilibrium point for flow fuzzy 𝜑
F(𝑈) → F(𝑈) is obtained by Zadeh’s extension 𝑥𝑒 : 𝑈 →
𝑈. Let 𝑥𝑒(𝑖) (𝑥𝑜 ) be an 𝑖th coordinated of equilibrium point 𝑥𝑒 .
Similarly, we can prove that 𝑖th projected of the equilibrium
point fuzzy x𝑒 = 𝑥̂𝑒 (x𝑜 ) ∈ F(𝑈) is the point x𝑖 = 𝑥̂𝑒(𝑖) (x𝑜 ) ∈
F(R) where 𝑥̂𝑒(𝑖) : F(𝑈) → F(R) is a Zadeh’s extension of
𝑥𝑒(𝑖) : 𝑈 → R. More briefly, for x𝑜 ∈ F(𝑈), the equality holds
following:
= [x𝑖 ] for all 𝛼 ∈ [0, 1] and the assertion
The solution of the model 𝑆𝐼, defined by functions
𝜑(1)
𝑡 (𝑆𝑜 , 𝐼𝑜 ) = 𝑁𝑜 (1 −
𝜑(2)
𝑡
𝐼𝑜
),
𝐼𝑜 + 𝑆𝑜 𝑒𝑁𝑜 𝑟𝑡
−𝑁𝑜 𝐼𝑜
(𝑆𝑜 , 𝐼𝑜 ) =
,
𝐼𝑜 + 𝑆𝑜 𝑒−𝑁𝑜 𝑟𝑡
(51)
6
Journal of Applied Mathematics
4
90
3
80
1
70
0
60
−1
𝜑t(1) (xo )
𝜑t(1) (xo )
2
−2
−3
−4
0
2
4
6
8
10
40
30
12
20
t
4
10
3
0
2
𝜑t(2) (xo )
50
0
1
2
3
4
1
6
7
8
9
10
6
7
8
9
10
t
0
100
−1
90
−2
80
−3
70
0
2
4
6
8
10
12
t
(2)
Figure 3: Time course of 𝜑(1)
𝑡 (x𝑜 ) and 𝜑𝑡 (x𝑜 ), respectively.
𝜑t(2) (xo )
−4
5
60
50
40
30
converges to the equilibrium point 𝑥e (𝑆𝑜 , 𝐼𝑜 ) = (0, 𝑆𝑜 + 𝐼𝑜 ).
According to what is discussed in [3], for all x𝑜 ∈ F(R2+ ),
̂ converges to the equilibrium point fuzzy
the fuzzy solution 𝜑
x𝑒 = 𝑥̂𝑒 (x𝑜 ).
According to the equality (46), projections of the equilibrium point x𝑒 on the coordinate axis are obtained by
extension of Zadeh components 𝑥𝑒 . That is, the projections
are fuzzy, respectively, x1 = 𝜒{0} and x, whose membership
function is given by
𝜇x2 (𝐼) = sup 𝜇x𝑜 (𝑆𝑜 , 𝐼𝑜 ) = sup 𝜇x𝑜 (𝑆𝑜 , 𝐼 − 𝑆𝑜 ) .
𝑆𝑜 +𝐼𝑜 =𝐼
(52)
𝑆𝑜
By Proposition 5, fuzzy projections of fuzzy solution
̂ 𝑡 (x𝑜 ), on F(R), of model 𝑆𝐼 are obtained by extension of
𝜑
(2)
Zadeh, the components 𝜑(1)
𝑡 and 𝜑𝑡 , given by
𝜑(1)
𝑡 (𝑆𝑜 , 𝐼𝑜 ) = 𝑁𝑜 (1 −
𝜑(2)
𝑡
𝐼𝑜
),
𝐼𝑜 + 𝑆𝑜 𝑒𝑁𝑜 𝑟𝑡
(53)
𝑁𝑜 𝐼𝑜
(𝑆𝑜 , 𝐼𝑜 ) =
.
𝐼𝑜 + 𝑆𝑜 𝑒𝑁𝑜 𝑟𝑡
To illustrate, suppose the force infection is 𝑟 = 0.01, and we
take the initial condition x𝑜 ∈ F(R2+ ) defined by membership
function
2
2
𝜇x𝑜 (𝑆𝑜 , 𝐼𝑜 ) = max {1 − 0.01(𝑆𝑜 − 80) − 0.25(𝐼𝑜 − 5) , 0} .
(54)
̂ (1)
Figure 4 shows the evolution of applications 𝜑
𝑡 (x𝑜 ) and
(2)
(1)
̂ 𝑡 (x𝑜 ) converges
̂ 𝑡 (x𝑜 ) with the time evolution. Note that 𝜑
𝜑
̂ (2)
to x1 = 𝜒{0} , whereas 𝜑
𝑡 (x𝑜 ) converges to x2 with the
membership function given by (52).
20
10
0
0
1
2
3
4
5
t
̂ 𝑡 (x𝑜 ) on the axes
Figure 4: Time evolution of the fuzzy projection 𝜑
𝑆 and 𝐼, respectively.
We also consider that the number of individuals in the
population is known, say 𝑁. In this case, the variables 𝑆
and 𝑅 are related by equality 𝑆 + 𝐼 = 𝑁. Under this
assumption, the deterministic solution converges to the point
of equilibrium 𝑥𝑒 (𝑆𝑜 , 𝐼𝑜 ) = (0, 𝑁), and, therefore, the fuzzy
solution converge to the equilibrium point fuzzy 𝜒{(0,𝑁)} . In
̂ (1)
̂ (2)
this case, the projections 𝜑
𝑡 (x𝑜 ) and 𝜑
𝑡 (x𝑜 ) converges to
𝜒{0} and 𝜒{𝑁} , respectively.
In Figure 5, we plot the projections of the fuzzy solution
𝜑𝑡 (x𝑜 ), to the initial condition 𝐼𝑜 = 20 and 𝑆𝑜 given by fuzzy
set
𝜇x𝑜 (𝑆𝑜 , 𝐼𝑜 )
2
={
max {1 − 0.01(𝑆𝑜 − 80) , 0} ,
0
if 𝑆𝑜 + 𝐼𝑜 = 𝑁, (55)
if 𝑆𝑜 + 𝐼𝑜 ≠ 𝑁.
The graphical representation of fuzzy projections of this
work is established as follows: given an 𝛼 ∈ [0, 1], the region
̂ (𝑖)
in plan bounded by 𝛼-level 𝜑
[0,𝑇] (x𝑜 ) is filled with a shade
̂ (𝑖)
of gray. If 𝛼 = 0, then the region bounded by 𝜑
[0,𝑇] (x𝑜 ) is
filled with the white color, whereas if 𝛼 = 1, then the region
Journal of Applied Mathematics
7
̂ 𝑡 : F(𝑈×𝑃) → F(𝑈) of 𝜑𝑡 (𝑥𝑜 , 𝑝) is well defined,
extension 𝜑
and according to (5), for all y𝑜 ∈ F(𝑈 × 𝑃) we have:
̂ t(1) (xo )
𝜑
80
𝛼
[̂
𝜑𝑡 (y𝑜 )] = 𝜑𝑡 ([y𝑜 ] ) .
40
From the standpoint of applications, it is important to
know the flow behavior of the deterministic phase space 𝑈 ⊂
R𝑛 of (56) instead of space 𝑈 × 𝑃 ⊂ R𝑛+𝑚 to (57), since the
flow components 𝜓𝑡 : 𝑈 × 𝑃 → 𝑈 × 𝑃, that are 𝑃 ⊂ R𝑚 , do
not have any additional information. It is worth noting that,
for all 𝑦𝑜 = (𝑥𝑜 , 𝑝𝑜 ), we have
20
0
0
2
4
6
t
100
𝑃𝑛 (𝜓𝑡 (𝑦𝑜 )) = 𝑃𝑛 (𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) , 𝑝𝑜 ) = 𝜑𝑡 (𝑦𝑜 ) .
80
̂ t(2) (xo )
𝜑
𝛼
60
40
20
2
4
6
t
̂ 𝑡 (x𝑜 ) on the axes
Figure 5: Time evolution of the fuzzy projection 𝜑
𝑆 and 𝐼 respectively.
̂ (𝑖)
bounded by 𝜑
[0,𝑇] (x𝑜 ) is filled with black. Thus, the larger the
degree of membership of a point 𝑥, the darker its color.
4. Parameters and Initial Condition Fuzzy
In [2] the problem of uncertainty in the parameters of a given
autonomous equation is solved using the strategy to consider
such parameters as the initial condition of an equation with
dimension higher than the original. More precisely, given an
autonomous equation that depends on a parameter vector
𝑝𝑜 ∈ 𝑃 ⊂ R𝑚
𝑑𝑥
= 𝑓 (𝑥, 𝑝𝑜 ) ,
𝑑𝑡
𝑥 (0) = 𝑥𝑜
(56)
̂ 𝑡 : F(𝑈 × 𝑃) → F(𝑈),
Proposition 8. The application 𝜑
given by Zadeh’s extension 𝜑𝑡 : 𝑈 × 𝑃 → 𝑈, is the fuzzy
̂ 𝑡 : F(𝑈 × 𝑃) → F(𝑈 × 𝑃) on
projection of fuzzy flow 𝜓
F(𝑈).
Proof. Let y𝑜 ∈ F(𝑈 × 𝑃) and fix 𝑡 ≥ 0. To prove the claim,
̂ 𝑡 (y𝑜 ) of
it suffices to show that y is the fuzzy projection of 𝜓
̂ 𝑡 (y𝑜 ).
F(𝑈), then y = 𝜑
To simplify, let Im (𝜑𝑡 ) be the image set of 𝜑𝑡 : 𝑈 × 𝑃 →
̂ 𝑡 (y𝑜 ) is given
𝑈. By definition, the membership function of 𝜑
by
sup 𝜇 (𝑥 , 𝑝 ) if 𝑥 ∈ Im (𝜑𝑡 ) ,
{
{ (𝑥 ,𝑝 ) y𝑜 𝑜 𝑜
𝜇𝜑̂ (y𝑜 ) (𝑥) = {𝜑𝑡 (𝑥𝑜𝑜,𝑝𝑜𝑜)=𝑥
(60)
𝑡
{
if 𝑥 ∉ Im (𝜑𝑡 ) .
{0
̂ 𝑡 (y𝑜 ) on F(𝑈). By
Let y ∈ F(𝑈) be the projection of 𝜓
definition of fuzzy projection, the membership function of
y ∈ F(𝑈) is given by
𝜇y (𝑥) = sup 𝜇𝜓̂ (y𝑜 ) (𝑥, 𝑝) .
define the equation,
𝑝∈𝑃
𝑑𝑥
= 𝑓 (𝑥, 𝑝) ,
𝑑𝑡
𝑑𝑝
= 0,
𝑑𝑡
𝑥 (0) = 𝑥𝑜 ,
(57)
𝑝 (0) = 𝑝𝑜 ,
𝑚
(59)
Analogously to the deterministic case, we can also be
̂ 𝑡 : F(𝑈 × 𝑃) →
interested only in the fuzzy flow behavior 𝜓
F(𝑈 × 𝑃) on the phase space F(𝑈). The fuzzy projections
defined at the outset of this work can then be used to obtain
̂ 𝑡 on the space F(𝑈).
the fuzzy flow behavior 𝜓
The following statement characterizes the relationship
̂ 𝑡 on the space F(𝑈) and
between the projection of fuzzy 𝜓
̂ 𝑡 : F(𝑈 × 𝑃) → F(𝑈) is a solution of
Zadeh’s extension: 𝜑
(56).
60
0
(58)
and thus, the parameter vector 𝑝𝑜 ∈ 𝑃 ⊂ R now is a part
̂ 𝑡 : F(𝑈 ×
of the initial condition. Thus, Zadeh’s extension 𝜓
𝑃) → F(𝑈×𝑃) to the flow 𝜓𝑡 : 𝑈×𝑃 → 𝑈×𝑃 generated by
(57) incorporates the uncertainties of initial conditions and
parameters of (56).
Once the solution 𝜑𝑡 : 𝑈 × 𝑃 → 𝑈 generated by (56)
is continuous in the initial condition and parameters, Zadeh’s
𝑡
(61)
Now, as 𝜓𝑡 (𝑥𝑜 , 𝑝𝑜 ) = (𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ), 𝑝𝑜 ), so, 𝑥 ∈ Im (𝜑𝑡 ) if
and only if (𝑥, 𝑝) ∈ Im (𝜓𝑡 ) for some 𝑝 ∈ 𝑃. So, for all
̂ 𝑡 (y𝑜 ) is
𝑥 ∈ Im (𝜑𝑡 ), the membership function of 𝜓
𝜇𝜓̂ (y𝑜 ) (𝑥, 𝑝) =
𝑡
=
=
sup
𝜓𝑡 (𝑥𝑜 ,𝑦)=(𝑥,𝑝)
𝜇y𝑜 (𝑥𝑜 , 𝑦)
sup 𝜇y𝑜 (𝑥𝑜 , 𝑦)
𝜑𝑡 (𝑥𝑜 ,𝑦)=𝑥
𝑦=𝑝
sup
𝑥𝑜 ∈𝑈
𝜑𝑡 (𝑥𝑜 ,𝑝)=𝑥
𝜇y𝑜 (𝑥𝑜 , 𝑝) .
(62)
8
Journal of Applied Mathematics
2.2
2
1.8
1.6
1.4
k 1.2
1
0.8
0.6
0.4
0.2
t = 1.8
t = 0.8
for all y𝑜 ∈ F(𝑈 × 𝑃) and 𝑡 ∈ R+ . Using the continuity of
applications 𝑃𝑛 and 𝜓𝑡 , we have
t=0
̂ 𝑡 (y𝑜 ))]
[𝑃̂𝑛 (𝜓
x
𝛼
𝛼
̂ 𝑡 ([y𝑜 ] )) = {𝑃𝑛 (𝜓
̂ 𝑡 (𝑦𝑜 )) : 𝑦𝑜 ∈ [x𝑜 ] }
= 𝑃𝑛 (𝜓
𝛼
= {𝑃𝑛 (𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] }
1.5
2
2.5
3
3.5
4
x
(a)
4
3.5
3
2.5
2
1.5
1
0.5
0
1
0
2
3
4
5
6
7
= {𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] }
𝛼
= 𝜑𝑡 ([y𝑜 ] ) ,
for all 𝛼 ∈ [0, 1]. The previous equality concludes the proof
proposition.
In contrast to [6, 7], when the equation depends on
parameters such as (56), the fuzzy solution proposed by fuzzy
Buckley and Feuring in [8] is obtained by Zadeh’s extension
flow deterministic 𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ). This way, Proposition 8 ensures
that the solution of fuzzy Buckley and Feuring is the fuzzy
projection of the fuzzy solution proposed by [6, 7].
Consider that subjective parameters in (56) contributes
to an increase in uncertainty. Set a parameter 𝑝 ∈ 𝑃, and
given a fuzzy initial condition x𝑜 , the 𝛼-levels to the fuzzy flow
generated by (56) are the sets
t
𝛼
𝛼
[̂
𝜑𝑡 (x𝑜 )] = {𝜑𝑡 (𝑥𝑜 , 𝑝) : 𝑥𝑜 ∈ [x𝑜 ] } .
(b)
̂ 𝑡 (y𝑜 ). (b) Evolution of fuzzy projection
Figure 6: (a) Evolution of 𝜓
̂ 𝑡 (y𝑜 ).
of 𝜑
If 𝑥 ∉ Im (𝜑𝑡 ), so, (𝑥, 𝑝) ∉ Im (𝜓𝑡 ) for all 𝑝 ∈ 𝑃, and so,
𝜇𝜓̂ (y𝑜 ) (𝑥, 𝑝) = 0.
𝑡
̂ 𝑡 (y𝑜 ) on F(𝑈) has the
But the fuzzy projection of 𝜓
membership function
sup 𝜇𝜓̂ (y𝑜 ) (𝑥, 𝑝) = sup
𝑡
𝑝∈𝑃
=
sup
𝑥𝑜 ∈𝑈
𝜑𝑡 (𝑥𝑜 ,𝑝)=𝑥
sup
(𝑥𝑜 ,𝑝)
𝜑𝑡 (𝑥𝑜 ,𝑝)=𝑥
𝑡
sup
(𝑥𝑜 ,𝑝)
𝜑𝑡 (𝑥𝑜 ,𝑝)=𝑥
𝜇y𝑜 (𝑥𝑜 , 𝑝) .
(64)
So, for all 𝑥 ∈ 𝑈, the value equality is as follows:
𝜇y (𝑥) = sup 𝜇𝜓̂ (y𝑜 ) (𝑥, 𝑝) = 𝜇𝜑̂ (y𝑜 ) (𝑥) ,
𝑝∈𝑃
𝑡
𝑡
𝛼
So, we have
𝛼
𝛼
[̂
𝜑𝑡 (x𝑜 , p𝑜 )] .
𝜑𝑡 (x𝑜 )] ⊆ [̂
(70)
𝑑𝑥
= 𝛽 (𝑘𝑜 − 𝑥)
𝑑𝑡
(63)
̂ 𝑡 (y𝑜 ) ∈ F(𝑈) has the
Now, by definition, the point 𝜑
membership function
𝜇𝜑̂ (y𝑜 ) (𝑥) =
𝛼
[̂
𝜑𝑡 (x𝑜 , p𝑜 )] = {𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [x𝑜 ] × [p𝑜 ] } .
(69)
Example 9. Consider the case where the parameter 𝑘𝑜 in the
equation
𝜇y𝑜 (𝑥𝑜 , 𝑝)
𝜇y𝑜 (𝑥𝑜 , 𝑝) .
(68)
On the other hand, if the 𝛼-levels of p𝑜 ∈ F(𝑃) contain 𝑝, so,
by Proposition 8, we have
𝛼
𝑝∈𝑃
(67)
𝛼
t=3
0.5
1
0
̂ t (xo )
𝜑
𝛼
(65)
(71)
is a fuzzy parameter. In the previous equation, the solution
𝜑𝑡 : R2 → R, in terms of 𝑥𝑜 and 𝑘𝑜 , is given by
𝜑𝑡 (𝑥𝑜 , 𝑘𝑜 ) = 𝑘𝑜 + (𝑥𝑜 − 𝑘𝑜 ) 𝑒−𝛽𝑡 ,
(72)
and thus the flow 2-dimensional 𝜓𝑡 : R2 → R2 , for the
case in which the parameter is incorporated into the initial
condition, is given by
𝜓𝑡 (𝑥𝑜 , 𝑘𝑜 ) = (𝑘𝑜 + (𝑥𝑜 − 𝑘𝑜 ) 𝑒−𝛽𝑡 , 𝑘𝑜 ) .
(73)
The proof of the proposition can also be made through
the 𝛼-levels. In fact, we must show that
̂ 𝑡 : F(R2 ) →
According to Proposition 8, Zadeh’s extension 𝜑
̂𝑡 :
F(R) of 𝜑𝑡 is the projection of F(R) fuzzy flow 𝜓
2
2
2
F(R ) → F(R ). To illustrate, consider y𝑜 ∈ F(R ). By
definition, we have
̂ 𝑡 (y𝑜 ))
̂ 𝑡 (y𝑜 ) = 𝑃̂𝑛 (𝜓
𝜑
̂ 𝑡 ([y𝑜 ] ) = {𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] } .
𝜑
which proves the assertion.
(66)
𝛼
𝛼
(74)
Journal of Applied Mathematics
9
̂ 𝑡 (y𝑜 )) has 𝛼-levels given by
Moreover, the projection 𝑃̂1 (𝜓
̂ 𝑡 (y𝑜 ))]
[𝑃̂𝑛 (𝜓
𝛼
𝛼
= 𝑃1 (𝜓𝑡 ([y𝑜 ] ))
𝛼
= {𝑃1 (𝜓𝑡 (𝑥𝑜 , 𝑝𝑜 )) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] }
(75)
𝛼
= {𝑃1 (𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] }
𝛼
= {𝜑𝑡 (𝑥𝑜 , 𝑝𝑜 ) : (𝑥𝑜 , 𝑝𝑜 ) ∈ [y𝑜 ] } ,
̂ 𝑡 (y𝑜 ))]𝛼 = 𝜑
̂ 𝑡 ([y𝑜 ]𝛼 ), and
from which we conclude that [𝑃̂𝑛 (𝜓
consequently,
̂ 𝑡 (y𝑜 )) = 𝜑
̂ 𝑡 (y𝑜 ) .
𝑃̂1 (𝜓
(76)
For any initial condition y𝑜 ∈ F(R2 ), we show that
̂ 𝑡 converges to the equilibrium points y𝑒 which is Zadeh’s
𝜓
extension 𝑦𝑒 : R2 → R2 given by 𝑦𝑒 (𝑥𝑜 , 𝑘𝑜 ) = (𝑘𝑜 , 𝑘𝑜 ). That
is, the equilibrium point y𝑒 has membership function
𝜇y𝑒 (𝑥, 𝑘) = min {𝜒{𝑘} (𝑥) , sup 𝜇y𝑜 (𝑥𝑜 , 𝑘)} .
𝑥𝑜
(77)
In particular, if y𝑜 = (x𝑜 , k𝑜 ) is the fuzzy cartesian product of
x𝑜 and k𝑜 ∈ F(R), then the membership function in this case
is given by
𝜇y𝑒 (𝑥, 𝑘) = sup 𝜇y𝑜 (𝑥𝑜 , 𝑘) = sup Δ (𝜇x𝑜 (𝑥𝑜 ) , 𝜇k𝑜 (𝑥))
𝑥𝑜
𝑥𝑜
(78)
= 𝜇k𝑜 (𝑥)
when 𝑥 = 𝑘 and 𝜇y𝑒 (𝑥, 𝑘) = 0, when 𝑥 ≠ 𝑘.
The projection x ∈ F(R) for this equilibrium point has
membership function
𝜇x (𝑥) = sup 𝜇y𝑒 (𝑥, 𝑘) = 𝜇k𝑜 (𝑥) ,
𝑘∈R
(79)
and we have 𝑑∞ (̂
𝜑𝑡 (y𝑜 ), x) → 0 as 𝑡 → ∞.
In Figure 6, we have the graphical representation of the
̂ 𝑡 (y𝑜 ) and its fuzzy projection 𝜑
̂ 𝑡 (y𝑜 ).
fuzzy solution 𝜓
5. Conclusions
In this paper, we define the concept of fuzzy projections and
study some of its main properties, in addition to establishing
some results on projections of fuzzy differential equations. As
we have seen, different concepts of fuzzy solutions of differential equations are related by fuzzy projections. Importantly,
by means of fuzzy projections, we can analyze the evolution
of fuzzy solutions over time.
References
[1] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory
and Applications, World Scientific, Singapore, 1994.
[2] M. T. Mizukoshi, L. C. Barros, Y. Chalco-Cano, H. RománFlores, and R. C. Bassanezi, “Fuzzy differential equations and
the extension principle,” Information Sciences, vol. 177, no. 17, pp.
3627–3635, 2007.
[3] M. S. Cecconello, Sistemas dinamicos em espaços metricos
fuzzy—aplicacoes em biomatematica [Ph.D. thesis], IMECC;
UNICAMP, 2010.
[4] C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis,
Springer, New York, NY, USA, 3rd edition, 2005.
[5] L. C. Barros, R. C. Bassanezi, and P. A. Tonelli, “On the
continuity of the Zadeh’s extension,” in Proceedings of the 7th
IFSA World Congress, vol. 2, Praga, 1997.
[6] M. Oberguggenberger and S. Pittschmann, “Differential equations with fuzzy parameters,” Mathematical and Computer
Modelling of Dynamical Systems, vol. 5, no. 3, pp. 181–202, 1999.
[7] M. T. Mizukoshi, Estabilidade de sistemas dinamicos fuzzy
[Ph.D. thesis], IMECC; UNICAMP, 2004.
[8] J. J. Buckley and T. Feuring, “Fuzzy differential equations,” Fuzzy
Sets and Systems, vol. 110, no. 1, pp. 43–54, 2000.
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