About Projections of Solutions for Fuzzy Differential Equations

In this paper we propose the concept of fuzzy projections on subspaces of F(R), obtained from Zadeh’s extension of canonical projections in R, 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.


Introduction
Consider the set  ⊂ R  .Denote by F() the set formed by the fuzzy subsets of  whose subsets have support compacts in .Some properties for metrics F() can be found in [1].If  is a subset of , we will use the notation   to indicate a membership function for the fuzzy set called membership function or crisp of .
Consider the autonomous equation defined by where  :  ⊂ R  → R  is a sufficiently smooth function.
For each   ∈ , denote by   (  ) the deterministic solution (1) with initial condition   .Here we are assuming that the solution is defined for all  ∈ R + .The function   :  →  will be called deterministic flow.
To consider initial conditions with inaccuracies modeled by fuzzy sets [2], consider the proposed Zadeh's extension   , the application φ : F() → F(), which takes the fuzzy set x  ∈ F() and the fuzzy set φ (x  ).In the context of this paper we call the application φ of fuzzy flow.Given x  ∈ F(), we say φ (x  ) is a fuzzy solution to (1) whose initial condition is the fuzzy set x  .
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(R  ), obtained from Zadeh's extension defined canonical projections in R  , 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.

Projections in Fuzzy Metric Spaces
We restrict our analysis to the set F() whose elements are subsets of a fuzzy set  whose -levels are compact and nonempty subsets in .The fuzzy subsets that are F() will be denoted by bold lowercase letters to differentiate the elements .So x ∈ F() if and only if [x]  is compact and nonempty subset for all  ∈ [0, 1].
We can define a structure of metric spaces in F() by the Hausdorff metric for compact subsets of .Let K() be the set formed by nonempty compact subsets of the metric space (, ).Given two sets ,  in K(), the distance between them can be defined by dist (, ) = sup ∈ inf ∈  (, ) . ( The distance between sets defined above is a pseudometric to K() since dist (, ) = 0 if and only if  ⊆ , not necessarily equal value.However, Hausdorff distance between ,  ∈ K() defined by is a metric for all K().so that (K(),   ) is a metric space.It is also worth that (, ) is a complete metric space, so (K(),   ) is also a complete metric space [4].
Through the Hausdorff metric   , we can define a metric for all F().Here we denote it by  ∞ .Given two points u, k ∈ F(), the distance between u, k is defined by It is not difficult to show that the distance defined above satisfies the properties of a metric and thus (F(),  ∞ ) is a metric space.Nguyen's theorem provides an important link between levels image of fuzzy subsets and the image of his -levels by a function  :  ×  → .According to [5], if  ⊆ R  and  ⊆ R  and  :  →  is continuous, then Zadeh's extension f : F() → F() is well defined and is worth for all  ∈ [0, 1] and u ∈ F().

Projections Fuzzy. Consider the application 𝑃
Provided that R  can be characterized as a subset of R + by identifying it with the subset R  × {0}, then the application   can be seen as the projection of R + on the set R  .For this reason, we say that  is the projection in R  ; the point (, ) ∈ R + .
Notice that a point (, V) is in the image of   if and only if V = 0. Furthermore,   (, ) =  for all  ∈ R  .Thus, given a point z ∈ F(R + ), with membership function  z : R + → [0, 1], the image P (z), obtained by Zadeh's extension projection   , has the membership function The application P : F(R + ) → F(R  ), obtained by Zadeh's extension of   , that for each z ∈ F(R + ) associates the point P (z) ∈ F(R  ) can be seen as a projection of F(R + ) in F(R  ), as it can be identified with the subset F(R  ) ×  {0} .Similarly the projection   satisfies: Based on this, we can define the projection of fuzzy z ∈ F(R + ) in F(R  ) as the point x ∈ F(R  ) with a membership function We also consider the function   : R + → R  that for all (, ) ∈ R + associates the point   (, ) =  ∈ R  .In this case, the image of a point z ∈ F(R + ), with the membership function  z : R + → [0, 1], is a point y ∈ F(R  ) with the membership function which we call fuzzy projection z in F(R  ).Thus the application P : F(R + ) → F(R  ) can be viewed as a fuzzy projection Here are some examples.
The image of z by applying P , in this case, has a membership function: Since min { a (),  b (V)} ≤  a (), so, sup As b ∈ F(R  ), so V ∈ R  so that  b (V) = 1.So, the fuzzy projection x of z about F(R  ) has a membership function: In Figure 1, the membership functions of z ∈ F(R 2 ), defined from a and b ∈ F(R) and your fuzzy projection in F(R), respectively, can be seen.In this figure, With similar argument, we can show that b ∈ F(R  ) is a fuzzy projection of z in F(R  ).
We can also define x = (a, b) ∈ F(R + ) through the - product, that is, The projection of z in F(R  ) has a membership function: sup Moreover, the projection F(R  ) has a membership function: sup Similarly, we can show that fuzzy projections z = (a, b) in F(R  ) and F(R  ) for all - Δ are, respectively, a and b.First, for any - Δ, we have So, sup But the ultimate is reached if we take Then, the projection of z = (a, b) in F(R  ) has membership function for all - Δ.
Example 2. Consider z ∈ F(R 2 ) determined by membership function For this case, we have the fuzzy projections x and y on F(R), respectively, determined by In Figure 2 we can see the membership functions z and x, respectively.Proposition 3. Let x = P (x) and y = P (y), with x and y ∈ F(R + ).The distance between the fuzzy projections x and y is always limited by the distance between x and y.We can prove that dist

Proof. In fact, for all
The fuzzy projection p ∈ F(R  ) to a point p ∈ F(R + ) satisfies another important property of the projections.Namely, the projection p is the point that minimizes the distance between the point p ∈ F(R + ) and the set F(R  ), the latter set is considered as a subset of F(R + ).

Proposition 4. The fuzzy projection p in
Proof.First, let us note the abuse of notation in the statement.The term  ∞ (p, z) only makes sense because we can see Therefore, we have ).Thus, we can conclude that, for all q ∈ F(R  ),  ∞ (p, q) ≥  ∞ (p, p), which proves the assertion.
We can also define fuzzy projections z ∈ F( × ) in F() and F(), where  ⊂ R  and  ⊂ R  .In this case, the supremum in membership functions ( 8) and ( 9) is taken on the sets  and , respectively, and properties shown above metrics remain valid.
We can also consider the projection   : R  → R from a point  = ( 1 ,  2 , . . .,   ) ∈ R  in th coordinate axis; that is,   () =   .As shown before, the projection of Zadeh's extension   defines the application π : F(R  ) → F(R) that we call for the th fuzzy projection of F(R  ) on F(R).Thus, given a point x ∈ F(R), the th fuzzy projection of x on F(R) is a point x  with membership function given by Again, if x = (a 1 , a 2 , . . ., a  ) is defined by fuzzy Cartesian product, then th fuzzy projection of x ∈ F(R  ) in F(R) is a point a  .For simplicity, consider x ∈ R 3 defined by By the properties of -, it follows that for all , ,  ∈ R. Thus, the second fuzzy projection x on F(R) is the point x 2 where the membership function is For the previous inequality, we have Taking  and  such that  a 1 () =  a 3 () = 1, equality is attained in the supremum, and hence, Induction proves the general case in which x ∈ F(R  ).
Through expression (8), we can determine the -levels of fuzzy projection x ∈ F(R  ) to a point z ∈ F(R + ).Indeed, if  x () ≥ , so,  ∈ R  such that  z (, ) ≥  so that (, ) ∈ [z]  .The reciprocal is also true, because if  z (, ) ≥ , then by (8),  x () ≥ .Thus, we conclude that: or Since applying   is continuous, we can use the equality (5) to show that the th fuzzy projection x  ∈ F(R) of x ∈ F(R  ) has -levels:

Projection of Fuzzy Solutions
where  ()  :  → R is the projection of the deterministic flow th coordinate axis; that is,  ()   (  ) is the th solution component   (  ), or even  ()   (  ) is the solution of the equation By applying Zadeh's extension to  ()  , we have the application φ()  : F() → F(R) that for each x  ∈ F() associates the image φ()  (x  ) ∈ F(R).As in the deterministic case, we show that the application φ () : F() → F(R) ia an th fuzzy projection to fuzzy flow φ : F() → F() on F(R).
We showed in [3] that the equilibrium point   deterministic flow   :  →  depends on the initial condition   ∈ ; then the equilibrium point for flow fuzzy φ : F() → F() is obtained by Zadeh's extension   :  → .Let  ()   (  ) be an th coordinated of equilibrium point   .Similarly, we can prove that th projected of the equilibrium point fuzzy x  = x (x  ) ∈ F() is the point x  = x()  (x  ) ∈ F(R) where x()  : F() → F(R) is a Zadeh's extension of  ()   :  → R.More briefly, for x  ∈ F(), the equality holds following: where x  is th fuzzy projection of the fuzzy equilibrium point x  .Consider just a few examples of the results presented previously.
Figure 3 shows the time evolution of the fuzzy projection of φ (x  ) on  and , respectively.Take the initial condition  0 defined by the membership function.
The solution of the model , defined by functions Figure 3: Time course of  (1)   (x  ) and  (2)   (x  ), respectively.
According to what is discussed in [3], for all x  ∈ F(R 2 + ), the fuzzy solution φ converges to the equilibrium point fuzzy x  = x (x  ).
According to the equality (46), projections of the equilibrium point x  on the coordinate axis are obtained by extension of Zadeh components   .That is, the projections are fuzzy, respectively, x 1 =  {0} and x, whose membership function is given by By Proposition 5, fuzzy projections of fuzzy solution φ (x  ), on F(R), of model  are obtained by extension of Zadeh, the components  (1)   and  (2)   , given by To illustrate, suppose the force infection is  = 0.01, and we take the initial condition x  ∈ F(R 2 + ) defined by membership function Figure 4 shows the evolution of applications φ(1)  (x  ) and φ(2) (x  ) with the time evolution.Note that φ(1)  (x  ) converges to x 1 =  {0} , whereas φ(2) (x  ) converges to x 2 with the membership function given by (52).We also consider that the number of individuals in the population is known, say .In this case, the variables  and  are related by equality  +  = .Under this assumption, the deterministic solution converges to the point of equilibrium   (  ,   ) = (0, ), and, therefore, the fuzzy solution converge to the equilibrium point fuzzy  {(0,)} .In this case, the projections φ(1) (x  ) and φ(2)  (x  ) converges to  {0} and  {} , respectively.
In Figure 5, we plot the projections of the fuzzy solution   (x  ), to the initial condition   = 20 and   given by fuzzy set The graphical representation of fuzzy projections of this work is established as follows: given an  ∈ [0, 1], the region in plan bounded by -level φ() [0,] (x  ) is filled with a shade of gray.If  = 0, then the region bounded by φ() [0,] (x  ) is filled with the white color, whereas if  = 1, then the region bounded by φ() [0,] (x  ) is filled with black.Thus, the larger the degree of membership of a point , the darker its color.
So, for all  ∈ , the value equality is as follows: which proves the assertion.
The proof of the proposition can also be made through the -levels.In fact, we must show that φ (y  ) = P (ψ  (y  )) for all y  ∈ F( × ) and  ∈ R + .Using the continuity of applications   and   , we have for all  ∈ [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   (  ,   ).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  ∈ , and given a fuzzy initial condition x  , the -levels to the fuzzy flow generated by (56) are the sets On the other hand, if the -levels of p  ∈ F() contain , so, by Proposition 8, we have So, we have Example 9. Consider the case where the parameter   in the equation is a fuzzy parameter.In the previous equation, the solution   : R 2 → R, in terms of   and   , is given by and thus the flow 2-dimensional   : R 2 → R 2 , for the case in which the parameter is incorporated into the initial condition, is given by   (  ,   ) = (  + (  −   )  − ,   ) .
For any initial condition y  ∈ F(R 2 ), we show that ψ converges to the equilibrium points y  which is Zadeh's extension   : R 2 → R 2 given by   (  ,   ) = (  ,   ).That is, the equilibrium point y  has membership function and we have  ∞ (φ  (y  ), x) → 0 as  → ∞.
In Figure 6, we have the graphical representation of the fuzzy solution ψ (y  ) and its fuzzy projection φ (y  ).

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.

1 𝜇Figure 1 :
Figure 1: Membership function of z and a respectively.

Figure 4 :
Figure 4: Time evolution of the fuzzy projection φ (x  ) on the axes  and , respectively.

Figure 5 :
Figure 5: Time evolution of the fuzzy projection φ (x  ) on the axes  and  respectively.