# About Projections of Solutions for Fuzzy Differential Equations

Authors: Moiseis S. Cecconello; Jefferson Leite; Rodney C. Bassanezi; Joao de Deus M. Silva
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

In this paper we propose the concept offuzzy projections on subspaces of ℱℝn, 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.


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## Body

## 1. Introduction

Consider the setU⊂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(1)dxdt=f(x),
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 φt, the application φ^t:ℱ(U)→ℱ(U), which takes the fuzzy set xo∈ℱ(U) and the fuzzy set φ^t(xo). In the context of this paper we call the application φ^t of fuzzy flow. Given xo∈ℱ(U), we say φ^t(xo) is a fuzzy solution to (1) whose initial 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 offuzzy 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 setℱ(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 inℱ(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 (X,d). Given two sets A,B in 𝒦(X), the distance between them can be defined by
(2)dist(A,B)=supa∈Ainfb∈Bd(a,b).The distance between sets defined above is apseudometric 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
(3)dH(A,B)=max{supa∈Ainfb∈Bd(a,b),supb∈Binfa∈Ad(a,b)}=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 metricdH, we can define a metric for all ℱ(X). Here we denote it by d∞. Given two points u,v∈ℱ(X), the distance between u, v is defined by
(4)d∞(u,v)=supα∈[0,1]dH([u]α,[v]α).
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 betweenα-levels 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 f^:ℱ(X)→ℱ(Y) is well defined and is worth
(5)[f^(u)]α=f([u]α)
for all α∈[0,1] and u∈ℱ(X).


### 2.1. Projections Fuzzy

Consider the applicationPn:ℝn+m→ℝn that for each (x,y)∈ℝn+m associates point Pn(x,y)=x∈ℝn.Provided thatℝn 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 P^n(z), obtained by Zadeh’s extension projection Pn, has the membership function
(6)μP^n(z)(x)=supv∈ℝmμz(x,v).The applicationP^n:ℱ(ℝn+m)→ℱ(ℝn), obtained by Zadeh’s extension of Pn, that for each z∈ℱ(ℝn+m) associates the point P^n(z)∈ℱ(ℝn) can be seen as a projection of ℱ(ℝn+m) in ℱ(ℝn), as it can be identified with the subset ℱ(ℝn)×χ{0}. Similarly the projection Pn satisfies:
(7)P^n(P^n(z))=P^n(z).Based on this, we can define theprojection of fuzzyz∈ℱ(ℝn+m) in ℱ(ℝn) as the point x∈ℱ(ℝn) with a membership function
(8)μx(x)=supa∈ℝmμz(x,a).We also consider the functionPm: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
(9)μy(y)=supa∈ℝnμz(a,y)
which we call fuzzy projectionz in ℱ(ℝm). Thus the application P^m:ℱ(ℝn+m)→ℱ(ℝm) can be viewed as a fuzzy projectionℱ(ℝn+m) in ℱ(ℝm).Here are some examples.Example 1.
Leta∈ℱ(ℝn) and b∈ℱ(ℝm). We can define z=(a,b)∈ℱ(ℝn+m) with membership function
(10)μz(x,y)=min{μa(x),μb(y)}.
The image ofz by applying P^n, in this case, has a membership function:
(11)μP^n(z)(x)=supv∈ℝmmin{μa(x),μb(v)}.
Sincemin{μa(x),μb(v)}≤μa(x), so,
(12)supv∈ℝmmin{μa(x),μb(v)}≤μa(x).
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:
(13)μx(x)=supv∈ℝmmin{μa(x),μb(v)}=μa(x).
In Figure1, 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,
(14)μa(x)=μb(x)=max{1-x2,0}.
With similar argument, we can show thatb∈ℱ(ℝm) is a fuzzy projection of z in ℱ(ℝm).
We can also definex=(a,b)∈ℱ(ℝn+m) through the t-norm product, that is,
(15)μz(x,y)=μa(x)μb(y).
The projection of z in ℱ(ℝn) has a membership function:
(16)supv∈ℝmμx(x,v)=supv∈ℝmμa(x)μb(v)=μa(x).
Moreover, the projection ℱ(ℝm) has a membership function:
(17)supu∈ℝnμx(u,y)=supu∈ℝnμa(u)μb(y)=μb(y).
Similarly, we can show that fuzzy projectionsz=(a,b) in ℱ(ℝn) and ℱ(ℝm) for all t-normΔ are, respectively, a and b. First, for any t-normΔ, we have
(18)Δ(μa(x),μb(y))≤Δ(μa(x),1)=μa(x).
So,
(19)supv∈ℝmΔ(μa(x),μb(v))≤μa(x).
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
(20)μx(x)=supv∈ℝmΔ(μa(x),μb(v))=μa(x),
for all t-normΔ.Membership function ofz and a respectively.Example 2.
Considerz∈ℱ(ℝ2) determined by membership function
(21)μz(x,y)=max{1-x2-2y2,0}.
For this case, we have the fuzzy projectionsx and y on ℱ(ℝ), respectively, determined by
(22)μx(x)=supv∈ℝmμz(x,v)=max{1-x2,0},μy(y)=supu∈ℝnμz(u,y)=max{1-2y2,0}.
In Figure2 we can see the membership functions z and x, respectively.Membership function ofz and x, respectively.Proposition 3.
Letx-=P^n(x) and y-=P^n(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 allα∈[0,1] we have
(23)dist([x]α,[y]α)=supa∈[x]αinfb∈[y]α∥a-b∥=sup(a1,a2)∈[x]αinf(b1,b2)∈[y]α∥a1-b1∥2+∥a2-b2∥2≥sup(a1,a2)∈[x]αinf(b1,b1)∈[y]α∥a1-b1∥2=supa1∈[x-]αinfb1∈[y-]α∥a1-b1∥2=dist([x-]α,[y-]α).
We can prove that dist([y]α,[x]α)≥dist([y-]α,[x-]α). Therefor,
(24)d∞(x-,y-)≤d∞(x,y).The fuzzy projectionp-∈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 projectionp- in ℱ(ℝn) of p∈ℱ(ℝn+m) is such that
(25)d∞(p,p-)=infz∈ℱ(ℝn)d∞(p,z).Proof.
First, let us note the abuse of notation in the statement. The termd∞(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
(26)∥x-y∥=∥x-y1∥2+∥y2∥2
since
(27)dist([p]α,[p-]α)=supy∈[p]αinfx∈[p-]α∥y1-x∥2+∥y2∥2=supy∈[p]α∥y2∥.
Moreover, we have(28)dist([p-]α,[p]α)=supx∈[p-]αinfy∈[p]α∥y1-x∥2+∥y2∥2.
Now, since x∈[p-]α, so, (x,z)∈[p]α for some z∈ℝm, where we have the inequality
(29)dist([p-]α,[p]α)=supx∈[p-]αinfy∈[p]α∥y1-x∥2+∥y2∥2≤∥z∥≤supy∈[p]α∥y2∥.
Thus, the Hausdorff distance between[p]α and [p-]α in this case is
(30)dH([p]α,[p-]α)=max{dist([p]α,[p-]α),dist([p-]α,[p]α)}=dist([p]α,[p-]α).
Letq∈ℱ(ℝn) such that q≠p-. This implies that [q]α≠[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
(31)∥x-y1∥2+∥y2∥2>∥y2∥
for all x∈[q]α. The second property follows directly from the projection inequality
(32)∥z1-y∥=∥z1-y1∥2+∥y2∥2>∥y2∥,
for all y=(y1,y2)∈[p]α. Thus in both cases we have to
(33)dist([p]α,[q]α)=supy∈[p]αinfx∈[p-]α∥y1-x∥2+∥y2∥2≥supy∈[p]α∥y2∥=dist([p]α,[p-]α).
Therefore, we havedH([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.We can also define fuzzy projectionsz∈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 projectionπi: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 extension πi defines the application π^i:ℱ(ℝn)→ℱ(ℝ) that we call for the ithfuzzy 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
(34)μxi(a)=supx∈ℝnxi=aμx(x).Again, ifx=(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
(35)μx(x,y,z)=Δ(Δ(μa1(x),μa2(y)),μa3(z)).
By the properties of t-norm, it follows that
(36)Δ(Δ(μa1(x),μa2(y)),μa3(z))≤Δ(Δ(μa1(x),μa2(y)),1)=Δ(μa1(x),μa2(y))≤μa2(y),
for all x,y,z∈ℝ.Thus, the second fuzzy projectionx on ℱ(ℝ) is the point x2 where the membership function is
(37)μx2(a)=supx∈ℝ3x2=aμx(x).
For the previous inequality, we have
(38)μx2(a)=supx∈ℝ3x2=aμx(x)≤μa2(a).
Taking x- and z- such that μa1(x-)=μa3(z-)=1, equality is attained in the supremum, and hence,
(39)μx2(a)=supx∈ℝ3x2=aμx(x)=μa2(a).Induction proves the general case in whichx∈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:
(40)x∈[x]α⟺(x,y)∈[z]αforsomey∈ℝm,
or
(41)[x]α={x∈ℝn:(x,y)∈[z]α}.Since applyingπi 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:
(42)[xi]α={a∈ℝ:x∈[x]α,xi=a}.


## 2.1. Projections Fuzzy

Consider the applicationPn:ℝn+m→ℝn that for each (x,y)∈ℝn+m associates point Pn(x,y)=x∈ℝn.Provided thatℝn 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 P^n(z), obtained by Zadeh’s extension projection Pn, has the membership function
(6)μP^n(z)(x)=supv∈ℝmμz(x,v).The applicationP^n:ℱ(ℝn+m)→ℱ(ℝn), obtained by Zadeh’s extension of Pn, that for each z∈ℱ(ℝn+m) associates the point P^n(z)∈ℱ(ℝn) can be seen as a projection of ℱ(ℝn+m) in ℱ(ℝn), as it can be identified with the subset ℱ(ℝn)×χ{0}. Similarly the projection Pn satisfies:
(7)P^n(P^n(z))=P^n(z).Based on this, we can define theprojection of fuzzyz∈ℱ(ℝn+m) in ℱ(ℝn) as the point x∈ℱ(ℝn) with a membership function
(8)μx(x)=supa∈ℝmμz(x,a).We also consider the functionPm: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
(9)μy(y)=supa∈ℝnμz(a,y)
which we call fuzzy projectionz in ℱ(ℝm). Thus the application P^m:ℱ(ℝn+m)→ℱ(ℝm) can be viewed as a fuzzy projectionℱ(ℝn+m) in ℱ(ℝm).Here are some examples.Example 1.
Leta∈ℱ(ℝn) and b∈ℱ(ℝm). We can define z=(a,b)∈ℱ(ℝn+m) with membership function
(10)μz(x,y)=min{μa(x),μb(y)}.
The image ofz by applying P^n, in this case, has a membership function:
(11)μP^n(z)(x)=supv∈ℝmmin{μa(x),μb(v)}.
Sincemin{μa(x),μb(v)}≤μa(x), so,
(12)supv∈ℝmmin{μa(x),μb(v)}≤μa(x).
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:
(13)μx(x)=supv∈ℝmmin{μa(x),μb(v)}=μa(x).
In Figure1, 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,
(14)μa(x)=μb(x)=max{1-x2,0}.
With similar argument, we can show thatb∈ℱ(ℝm) is a fuzzy projection of z in ℱ(ℝm).
We can also definex=(a,b)∈ℱ(ℝn+m) through the t-norm product, that is,
(15)μz(x,y)=μa(x)μb(y).
The projection of z in ℱ(ℝn) has a membership function:
(16)supv∈ℝmμx(x,v)=supv∈ℝmμa(x)μb(v)=μa(x).
Moreover, the projection ℱ(ℝm) has a membership function:
(17)supu∈ℝnμx(u,y)=supu∈ℝnμa(u)μb(y)=μb(y).
Similarly, we can show that fuzzy projectionsz=(a,b) in ℱ(ℝn) and ℱ(ℝm) for all t-normΔ are, respectively, a and b. First, for any t-normΔ, we have
(18)Δ(μa(x),μb(y))≤Δ(μa(x),1)=μa(x).
So,
(19)supv∈ℝmΔ(μa(x),μb(v))≤μa(x).
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
(20)μx(x)=supv∈ℝmΔ(μa(x),μb(v))=μa(x),
for all t-normΔ.Membership function ofz and a respectively.Example 2.
Considerz∈ℱ(ℝ2) determined by membership function
(21)μz(x,y)=max{1-x2-2y2,0}.
For this case, we have the fuzzy projectionsx and y on ℱ(ℝ), respectively, determined by
(22)μx(x)=supv∈ℝmμz(x,v)=max{1-x2,0},μy(y)=supu∈ℝnμz(u,y)=max{1-2y2,0}.
In Figure2 we can see the membership functions z and x, respectively.Membership function ofz and x, respectively.Proposition 3.
Letx-=P^n(x) and y-=P^n(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 allα∈[0,1] we have
(23)dist([x]α,[y]α)=supa∈[x]αinfb∈[y]α∥a-b∥=sup(a1,a2)∈[x]αinf(b1,b2)∈[y]α∥a1-b1∥2+∥a2-b2∥2≥sup(a1,a2)∈[x]αinf(b1,b1)∈[y]α∥a1-b1∥2=supa1∈[x-]αinfb1∈[y-]α∥a1-b1∥2=dist([x-]α,[y-]α).
We can prove that dist([y]α,[x]α)≥dist([y-]α,[x-]α). Therefor,
(24)d∞(x-,y-)≤d∞(x,y).The fuzzy projectionp-∈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 projectionp- in ℱ(ℝn) of p∈ℱ(ℝn+m) is such that
(25)d∞(p,p-)=infz∈ℱ(ℝn)d∞(p,z).Proof.
First, let us note the abuse of notation in the statement. The termd∞(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
(26)∥x-y∥=∥x-y1∥2+∥y2∥2
since
(27)dist([p]α,[p-]α)=supy∈[p]αinfx∈[p-]α∥y1-x∥2+∥y2∥2=supy∈[p]α∥y2∥.
Moreover, we have(28)dist([p-]α,[p]α)=supx∈[p-]αinfy∈[p]α∥y1-x∥2+∥y2∥2.
Now, since x∈[p-]α, so, (x,z)∈[p]α for some z∈ℝm, where we have the inequality
(29)dist([p-]α,[p]α)=supx∈[p-]αinfy∈[p]α∥y1-x∥2+∥y2∥2≤∥z∥≤supy∈[p]α∥y2∥.
Thus, the Hausdorff distance between[p]α and [p-]α in this case is
(30)dH([p]α,[p-]α)=max{dist([p]α,[p-]α),dist([p-]α,[p]α)}=dist([p]α,[p-]α).
Letq∈ℱ(ℝn) such that q≠p-. This implies that [q]α≠[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
(31)∥x-y1∥2+∥y2∥2>∥y2∥
for all x∈[q]α. The second property follows directly from the projection inequality
(32)∥z1-y∥=∥z1-y1∥2+∥y2∥2>∥y2∥,
for all y=(y1,y2)∈[p]α. Thus in both cases we have to
(33)dist([p]α,[q]α)=supy∈[p]αinfx∈[p-]α∥y1-x∥2+∥y2∥2≥supy∈[p]α∥y2∥=dist([p]α,[p-]α).
Therefore, we havedH([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.We can also define fuzzy projectionsz∈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 projectionπi: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 extension πi defines the application π^i:ℱ(ℝn)→ℱ(ℝ) that we call for the ithfuzzy 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
(34)μxi(a)=supx∈ℝnxi=aμx(x).Again, ifx=(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
(35)μx(x,y,z)=Δ(Δ(μa1(x),μa2(y)),μa3(z)).
By the properties of t-norm, it follows that
(36)Δ(Δ(μa1(x),μa2(y)),μa3(z))≤Δ(Δ(μa1(x),μa2(y)),1)=Δ(μa1(x),μa2(y))≤μa2(y),
for all x,y,z∈ℝ.Thus, the second fuzzy projectionx on ℱ(ℝ) is the point x2 where the membership function is
(37)μx2(a)=supx∈ℝ3x2=aμx(x).
For the previous inequality, we have
(38)μx2(a)=supx∈ℝ3x2=aμx(x)≤μa2(a).
Taking x- and z- such that μa1(x-)=μa3(z-)=1, equality is attained in the supremum, and hence,
(39)μx2(a)=supx∈ℝ3x2=aμx(x)=μa2(a).Induction proves the general case in whichx∈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:
(40)x∈[x]α⟺(x,y)∈[z]αforsomey∈ℝm,
or
(41)[x]α={x∈ℝn:(x,y)∈[z]α}.Since applyingπi 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:
(42)[xi]α={a∈ℝ:x∈[x]α,xi=a}.


## 3. Projection of Fuzzy Solutions

### 3.1. Projection on the Coordinate Axes

Consider the flowφt:U⊂ℝn→U generated by the autonomous equation
(43)dxdt=f(x),
where φt(i):U→ℝ is the projection of the deterministic flow ith coordinate axis; that is, φt(i)(xo) is the ith solution component φt(xo), or even φt(i)(xo) is the solution of the equation
(44)dxidt=fi(x),x(0)=xo.By applying Zadeh’s extension toφt(i), we have the application φ^t(i):ℱ(U)→ℱ(ℝ) that for each xo∈ℱ(U) associates the image φ^t(i)(xo)∈ℱ(ℝ). As in the deterministic case, we show that the application φ^t(i):ℱ(U)→ℱ(ℝ) ia an ith fuzzy projection to fuzzy flow φ^t:ℱ(U)→ℱ(U) on ℱ(ℝ).Proposition 5.
The applicationφ^t(i):ℱ(U)→ℱ(ℝ) is ith fuzzy projection of fuzzy flow φ^t:ℱ(U)→ℱ(U) on ℱ(ℝ).Proof.
Letxo∈ℱ(U). By definition, ith fuzzy projection φ^t(xo) on ℱ(ℝ) is the point xi=π^i(φ^t(xo)). Since the projection is a continuous map, then it is worth
(45)[xi]α=πi(φt([xo]α))={πi(φt(xo)):xo∈[xo]α}={φt(i)(xo):xo∈[xo]α}=φt(i)([xo]α).
Then,[φ^t(i)(xo)]α=[xi]α for all α∈[0,1] and the assertion is proved.We showed in [3] that the equilibrium point xe deterministic flow φt:U→U depends on the initial condition xo∈U; then the equilibrium point for flow fuzzy φ^t: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=x^e(xo)∈ℱ(U) is the point x-i=x^e(i)(xo)∈ℱ(ℝ) where x^e(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:
(46)μx-i(x)=μx^e(i)(xo)(x),
where x-i 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(47)dx1dt=x2,x1(0)=x01,dx2dt=-x1,x2(0)=x02,
determines the flow two-dimensional φt:ℝ2→ℝ2, φt=(φt(1),φt(2)), given by
(48)φt(1)(x01,x02)=x01cost+x02sent,φt(2)(x01,x02)=x02cost-x01sent.
We have already shown in [3] that the fuzzy solution φ^t(xo) this equation is periodic for any choice of initial condition xo∈ℱ(ℝ2). According to the previous proposition, projections of fuzzy φ^t:ℱ(ℝ2)→ℱ(ℝ2) on ℱ(ℝ) are obtained by taking extensions of components Zadeh φt(1) and φt(2).
Figure3 shows the time evolution of the fuzzy projection of φ^t(xo) on x and y, respectively. Take the initial condition x0 defined by the membership function. 									(49)μxo(x,y)=max{1-(x-3)2-y2,0}.Time course ofφt(1)(xo) and φt(2)(xo), respectively.Example 7.
Consider the epidemiological modelSI defined by equations
(50)dSdt=-rSI,S(0)=So>0,dIdt=rSI,I(0)=Io>0,S=I+N.
The solution of the modelSI, defined by functions
(51)φt(1)(So,Io)=No(1-IoIo+SoeNort),φt(2)(So,Io)=-NoIoIo+Soe-Nort,
converges to the equilibrium point xe(So,Io)=(0,So+Io). According to what is discussed in [3], for all xo∈ℱ(ℝ+2), the fuzzy solution φ^ converges to the equilibrium point fuzzy xe=x^e(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, x-1=χ{0} and x-, whose membership function is given by
(52)μx-2(I)=supSo+Io=Iμxo(So,Io)=supSoμxo(So,I-So).
By Proposition5, fuzzy projections of fuzzy solution φ^t(xo), on ℱ(ℝ), of model SI are obtained by extension of Zadeh, the components φt(1) and φt(2), given by
(53)φt(1)(So,Io)=No(1-IoIo+SoeNort),φt(2)(So,Io)=NoIoIo+SoeNort.
To illustrate, suppose theforce infection is r=0.01, and we take the initial condition xo∈ℱ(ℝ+2) defined by membership function
(54)μxo(So,Io)=max{1-0.01(So-80)2-0.25(Io-5)2,0}.
Figure4 shows the evolution of applications φ^t(1)(xo) and φ^t(2)(xo) with the time evolution. Note that φ^t(1)(xo) converges to x-1=χ{0}, whereas φ^t(2)(xo) converges to x-2 with the membership function given by (52).
We also consider that the number of individuals in the population is known, sayN. 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 this case, the projections φ^t(1)(xo) and φ^t(2)(xo) converges to χ{0} and χ{N}, respectively.
In Figure5, we plot the projections of the fuzzy solution φt(xo), to the initial condition Io=20 and So given by fuzzy set
(55)μxo(So,Io)={max{1-0.01(So-80)2,0},ifSo+Io=N,0ifSo+Io≠N.Time evolution of the fuzzy projectionφ^t(xo) on the axes S and I, respectively.Time evolution of the fuzzy projectionφ^t(xo) on the axes S and I respectively.The graphical representation of fuzzy projections of this work is established as follows: given anα∈[0,1], the region in plan bounded by α-level φ^[0,T](i)(xo) is filled with a shade of gray. If α=0, then the region bounded by φ^[0,T](i)(xo) is filled with the white color, whereas if α=1, then the region bounded by φ^[0,T](i)(xo) is filled with black. Thus, the larger the degree of membership of a point x, the darker its color.


## 3.1. Projection on the Coordinate Axes

Consider the flowφt:U⊂ℝn→U generated by the autonomous equation
(43)dxdt=f(x),
where φt(i):U→ℝ is the projection of the deterministic flow ith coordinate axis; that is, φt(i)(xo) is the ith solution component φt(xo), or even φt(i)(xo) is the solution of the equation
(44)dxidt=fi(x),x(0)=xo.By applying Zadeh’s extension toφt(i), we have the application φ^t(i):ℱ(U)→ℱ(ℝ) that for each xo∈ℱ(U) associates the image φ^t(i)(xo)∈ℱ(ℝ). As in the deterministic case, we show that the application φ^t(i):ℱ(U)→ℱ(ℝ) ia an ith fuzzy projection to fuzzy flow φ^t:ℱ(U)→ℱ(U) on ℱ(ℝ).Proposition 5.
The applicationφ^t(i):ℱ(U)→ℱ(ℝ) is ith fuzzy projection of fuzzy flow φ^t:ℱ(U)→ℱ(U) on ℱ(ℝ).Proof.
Letxo∈ℱ(U). By definition, ith fuzzy projection φ^t(xo) on ℱ(ℝ) is the point xi=π^i(φ^t(xo)). Since the projection is a continuous map, then it is worth
(45)[xi]α=πi(φt([xo]α))={πi(φt(xo)):xo∈[xo]α}={φt(i)(xo):xo∈[xo]α}=φt(i)([xo]α).
Then,[φ^t(i)(xo)]α=[xi]α for all α∈[0,1] and the assertion is proved.We showed in [3] that the equilibrium point xe deterministic flow φt:U→U depends on the initial condition xo∈U; then the equilibrium point for flow fuzzy φ^t: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=x^e(xo)∈ℱ(U) is the point x-i=x^e(i)(xo)∈ℱ(ℝ) where x^e(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:
(46)μx-i(x)=μx^e(i)(xo)(x),
where x-i 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(47)dx1dt=x2,x1(0)=x01,dx2dt=-x1,x2(0)=x02,
determines the flow two-dimensional φt:ℝ2→ℝ2, φt=(φt(1),φt(2)), given by
(48)φt(1)(x01,x02)=x01cost+x02sent,φt(2)(x01,x02)=x02cost-x01sent.
We have already shown in [3] that the fuzzy solution φ^t(xo) this equation is periodic for any choice of initial condition xo∈ℱ(ℝ2). According to the previous proposition, projections of fuzzy φ^t:ℱ(ℝ2)→ℱ(ℝ2) on ℱ(ℝ) are obtained by taking extensions of components Zadeh φt(1) and φt(2).
Figure3 shows the time evolution of the fuzzy projection of φ^t(xo) on x and y, respectively. Take the initial condition x0 defined by the membership function. 									(49)μxo(x,y)=max{1-(x-3)2-y2,0}.Time course ofφt(1)(xo) and φt(2)(xo), respectively.Example 7.
Consider the epidemiological modelSI defined by equations
(50)dSdt=-rSI,S(0)=So>0,dIdt=rSI,I(0)=Io>0,S=I+N.
The solution of the modelSI, defined by functions
(51)φt(1)(So,Io)=No(1-IoIo+SoeNort),φt(2)(So,Io)=-NoIoIo+Soe-Nort,
converges to the equilibrium point xe(So,Io)=(0,So+Io). According to what is discussed in [3], for all xo∈ℱ(ℝ+2), the fuzzy solution φ^ converges to the equilibrium point fuzzy xe=x^e(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, x-1=χ{0} and x-, whose membership function is given by
(52)μx-2(I)=supSo+Io=Iμxo(So,Io)=supSoμxo(So,I-So).
By Proposition5, fuzzy projections of fuzzy solution φ^t(xo), on ℱ(ℝ), of model SI are obtained by extension of Zadeh, the components φt(1) and φt(2), given by
(53)φt(1)(So,Io)=No(1-IoIo+SoeNort),φt(2)(So,Io)=NoIoIo+SoeNort.
To illustrate, suppose theforce infection is r=0.01, and we take the initial condition xo∈ℱ(ℝ+2) defined by membership function
(54)μxo(So,Io)=max{1-0.01(So-80)2-0.25(Io-5)2,0}.
Figure4 shows the evolution of applications φ^t(1)(xo) and φ^t(2)(xo) with the time evolution. Note that φ^t(1)(xo) converges to x-1=χ{0}, whereas φ^t(2)(xo) converges to x-2 with the membership function given by (52).
We also consider that the number of individuals in the population is known, sayN. 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 this case, the projections φ^t(1)(xo) and φ^t(2)(xo) converges to χ{0} and χ{N}, respectively.
In Figure5, we plot the projections of the fuzzy solution φt(xo), to the initial condition Io=20 and So given by fuzzy set
(55)μxo(So,Io)={max{1-0.01(So-80)2,0},ifSo+Io=N,0ifSo+Io≠N.Time evolution of the fuzzy projectionφ^t(xo) on the axes S and I, respectively.Time evolution of the fuzzy projectionφ^t(xo) on the axes S and I respectively.The graphical representation of fuzzy projections of this work is established as follows: given anα∈[0,1], the region in plan bounded by α-level φ^[0,T](i)(xo) is filled with a shade of gray. If α=0, then the region bounded by φ^[0,T](i)(xo) is filled with the white color, whereas if α=1, then the region bounded by φ^[0,T](i)(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(56)dxdt=f(x,po),x(0)=xo
define the equation,
(57)dxdt=f(x,p),x(0)=xo,dpdt=0,p(0)=po,
and thus, the parameter vector po∈P⊂ℝm now is a part of the initial condition. Thus, Zadeh’s extension ψ^t:ℱ(U×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 solutionφt:U×P→U generated by (56) is continuous in the initial condition and parameters, Zadeh's extension φ^t:ℱ(U×P)→ℱ(U) of φt(xo,p) is well defined, and according to (5), for all yo∈ℱ(U×P) we have:
(58)[φ^t(yo)]α=φt([yo]α).From the standpoint of applications, it is important to know the flow behavior of the deterministic phase spaceU⊂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
(59)Pn(ψt(yo))=Pn(φt(xo,po),po)=φt(yo).Analogously to the deterministic case, we can also be interested only in the fuzzy flow behaviorψ^t:ℱ(U×P)→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 the fuzzy flow behavior ψ^t on the space ℱ(U).The following statement characterizes the relationship between the projection of fuzzyψ^t on the space ℱ(U) and Zadeh's extension: φ^t:ℱ(U×P)→ℱ(U) is a solution of (56).Proposition 8.
The applicationφ^t:ℱ(U×P)→ℱ(U), given by Zadeh's extension φt:U×P→U, is the fuzzy projection of fuzzy flow ψ^t:ℱ(U×P)→ℱ(U×P) on ℱ(U).Proof.
Letyo∈ℱ(U×P) and fix t≥0. To prove the claim, it suffices to show that y is the fuzzy projection of ψ^t(yo) of ℱ(U), then y=φ^t(yo).
To simplify, letIm(φt) be the image set of φt:U×P→U. By definition, the membership function of φ^t(yo) is given by
(60)μφ^t(yo)(x)={sup(xo,po)φt(xo,po)=xμyo(xo,po)ifx∈Im(φt),0ifx∉Im(φt).
Lety∈ℱ(U) be the projection of ψ^t(yo) on ℱ(U). By definition of fuzzy projection, the membership function of y∈ℱ(U) is given by
(61)μy(x)=supp∈Pμψ^t(yo)(x,p).
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 x∈Im(φt), the membership function of ψ^t(yo) is
(62)μψ^t(yo)(x,p)=supψt(xo,y)=(x,p)μyo(xo,y)=supφt(xo,y)=xy=pμyo(xo,y)=supxo∈Uφt(xo,p)=xμyo(xo,p).
If x∉Im(φt), so, (x,p)∉Im(ψt) for all p∈P, and so, μψ^t(yo)(x,p)=0.
But the fuzzy projection ofψ^t(yo) on ℱ(U) has the membership function
(63)supp∈Pμψ^t(yo)(x,p)=supp∈Psupxo∈Uφt(xo,p)=xμyo(xo,p)=sup(xo,p)φt(xo,p)=xμyo(xo,p).
Now, by definition, the pointφ^t(yo)∈ℱ(U) has the membership function
(64)μφ^t(yo)(x)=sup(xo,p)φt(xo,p)=xμyo(xo,p).
So, for allx∈U, the value equality is as follows:
(65)μy(x)=supp∈Pμψ^t(yo)(x,p)=μφ^t(yo)(x),
which proves the assertion.The proof of the proposition can also be made through theα-levels. In fact, we must show that
(66)φ^t(yo)=P^n(ψ^t(yo))
for all yo∈ℱ(U×P) and t∈ℝ+. Using the continuity of applications Pn and ψt, we have
(67)[P^n(ψ^t(yo))]α=Pn(ψ^t([yo]α))={Pn(ψ^t(yo)):yo∈[xo]α}={Pn(φt(xo,po),po):(xo,po)∈[yo]α}={φt(xo,po):(xo,po)∈[yo]α}=φt([yo]α),
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
(68)[φ^t(xo)]α={φt(xo,p-):xo∈[xo]α}.
On the other hand, if the α-levels of po∈ℱ(P) contain p-, so, by Proposition 8, we have
(69)[φ^t(xo,po)]α={φt(xo,po):(xo,po)∈[xo]α×[po]α}.
So, we have
(70)[φ^t(xo)]α⊆[φ^t(xo,po)]α.Example 9.
Consider the case where the parameterko in the equation
(71)dxdt=β(ko-x)
is a fuzzy parameter. In the previous equation, the solution φt:ℝ2→ℝ, in terms of xo and ko, is given by
(72)φt(xo,ko)=ko+(xo-ko)e-βt,
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
(73)ψt(xo,ko)=(ko+(xo-ko)e-βt,ko).
According to Proposition 8, Zadeh's extension φ^t:ℱ(ℝ2)→ℱ(ℝ) of φt is the projection of ℱ(ℝ) fuzzy flow ψ^t:ℱ(ℝ2)→ℱ(ℝ2). To illustrate, consider yo∈ℱ(ℝ2). By definition, we have
(74)φ^t([yo]α)={φt(xo,po):(xo,po)∈[yo]α}.
Moreover, the projection P^1(ψ^t(yo)) has α-levels given by
(75)[P^n(ψ^t(yo))]α=P1(ψt([yo]α))={P1(ψt(xo,po)):(xo,po)∈[yo]α}={P1(φt(xo,po),po):(xo,po)∈[yo]α}={φt(xo,po):(xo,po)∈[yo]α},
from which we conclude that [P^n(ψ^t(yo))]α=φ^t([yo]α), and consequently,
(76)P^1(ψ^t(yo))=φ^t(yo).
For any initial conditionyo∈ℱ(ℝ2), we show that ψ^t converges to the equilibrium points ye which is Zadeh's extension ye:ℝ2→ℝ2 given by ye(xo,ko)=(ko,ko). That is, the equilibrium point ye has membership function
(77)μye(x,k)=min{χ{k}(x),supxoμyo(xo,k)}.
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
(78)μye(x,k)=supxoμyo(xo,k)=supxoΔ(μxo(xo),μko(x))=μko(x)
when x=k and μye(x,k)=0, when x≠k.
The projectionx-∈ℱ(ℝ) for this equilibrium point has membership function
(79)μx-(x)=supk∈ℝμye(x,k)=μko(x),
and we have d∞(φ^t(yo),x-)→0 as t→∞.
In Figure6, we have the graphical representation of the fuzzy solution ψ^t(yo) and its fuzzy projection φ^t(yo).(a) Evolution ofψ^t(yo). (b) Evolution of fuzzy projection of φ^t(yo).
(a)
(b)


## 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.



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*Source: 184950-2013-06-17.xml*