Abstract

In this paper we propose the concept of fuzzy projections on subspaces of , obtained from Zadeh's extension of canonical projections in , 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 . Denote by the set formed by the fuzzy subsets of whose subsets have support compacts in . Some properties for metrics 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 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 . 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 , which takes the fuzzy set and the fuzzy set . In the context of this paper we call the application of fuzzy flow. Given , we say is a fuzzy solution to (1) whose initial condition is the fuzzy set .

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 , obtained from Zadeh’s extension defined canonical projections in , 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 whose elements are subsets of a fuzzy set whose -levels are compact and nonempty subsets in . The fuzzy subsets that are will be denoted by bold lowercase letters to differentiate the elements . So if and only if is compact and nonempty subset for all .

We can define a structure of metric spaces in by the Hausdorff metric for compact subsets of . Let be the set formed by nonempty compact subsets of the metric space . Given two sets in , the distance between them can be defined by

The distance between sets defined above is a pseudometric to since if and only if , not necessarily equal value. However, Hausdorff distance between , defined by is a metric for all . so that is a metric space. It is also worth that is a complete metric space, so is also a complete metric space [4].

Through the Hausdorff metric , we can define a metric for all . Here we denote it by . Given two points , the distance between , is defined by It is not difficult to show that the distance defined above satisfies the properties of a metric and thus 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 and and is continuous, then Zadeh's extension is well defined and is worth for all and .

2.1. Projections Fuzzy

Consider the application that for each associates point .

Provided that can be characterized as a subset of by identifying it with the subset , then the application can be seen as the projection of on the set . For this reason, we say that is the projection in ; the point .

Notice that a point is in the image of if and only if . Furthermore, for all . Thus, given a point , with membership function , the image , obtained by Zadeh’s extension projection , has the membership function

The application , obtained by Zadeh’s extension of , that for each associates the point can be seen as a projection of in , as it can be identified with the subset . Similarly the projection satisfies:

Based on this, we can define the projection of fuzzy in as the point with a membership function

We also consider the function that for all associates the point . In this case, the image of a point , with the membership function , is a point with the membership function which we call fuzzy projection in . Thus the application can be viewed as a fuzzy projection in .

Here are some examples.

Example 1. Let and . We can define with membership function
The image of by applying , in this case, has a membership function:
Since , so, As , so so that . So, the fuzzy projection of about has a membership function:
In Figure 1, the membership functions of , defined from and and your fuzzy projection in , respectively, can be seen. In this figure,
With similar argument, we can show that is a fuzzy projection of in .
We can also define through the product, that is, The projection of in has a membership function: Moreover, the projection has a membership function:
Similarly, we can show that fuzzy projections in and for all are, respectively, and . First, for any , we have So, But the ultimate is reached if we take so that . Then, the projection of in has membership function for all .

Example 2. Consider determined by membership function
For this case, we have the fuzzy projections and on , respectively, determined by
In Figure 2 we can see the membership functions and , respectively.

Proposition 3. Let and , with and . The distance between the fuzzy projections and is always limited by the distance between and .

Proof. In fact, for all we have We can prove that . Therefor,

The fuzzy projection to a point satisfies another important property of the projections. Namely, the projection is the point that minimizes the distance between the point and the set , the latter set is considered as a subset of .

Proposition 4. The fuzzy projection in of is such that

Proof. First, let us note the abuse of notation in the statement. The term only makes sense because we can see as a subset of . Provided that and , for and , we have since
Moreover, we have Now, since , so, for some , where we have the inequality
Thus, the Hausdorff distance between and in this case is
Let such that . This implies that , for some . Consequently, there such that or exists such that , for all . Namely, . For the first case, we have for all . The second property follows directly from the projection inequality for all . Thus in both cases we have to
Therefore, we have . Thus, we can conclude that, for all , which proves the assertion.

We can also define fuzzy projections in and , where and . 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 from a point in th coordinate axis; that is, . As shown before, the projection of Zadeh’s extension defines the application that we call for the th fuzzy projection of on . Thus, given a point , the th fuzzy projection of on is a point with membership function given by

Again, if is defined by fuzzy Cartesian product, then th fuzzy projection of in is a point . For simplicity, consider defined by By the properties of , it follows that for all .

Thus, the second fuzzy projection on is the point where the membership function is For the previous inequality, we have Taking and such that , equality is attained in the supremum, and hence,

Induction proves the general case in which .

Through expression (8), we can determine the -levels of fuzzy projection to a point . Indeed, if , so, such that so that . The reciprocal is also true, because if , then by (8), . Thus, we conclude that: or

Since applying is continuous, we can use the equality (5) to show that the th fuzzy projection of has -levels:

3. Projection of Fuzzy Solutions

3.1. Projection on the Coordinate Axes

Consider the flow generated by the autonomous equation where 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 that for each associates the image . As in the deterministic case, we show that the application ia an th fuzzy projection to fuzzy flow on .

Proposition 5. The application is th fuzzy projection of fuzzy flow on .

Proof. Let . By definition, th fuzzy projection on is the point . Since the projection is a continuous map, then it is worth
Then, for all and the assertion is proved.

We showed in [3] that the equilibrium point deterministic flow depends on the initial condition ; then the equilibrium point for flow fuzzy 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 is the point where is a Zadeh’s extension of . More briefly, for , the equality holds following: where is th fuzzy projection of the fuzzy equilibrium point .

Consider just a few examples of the results presented previously.

Example 6. The autonomous equation determines the flow two-dimensional , , given by
We have already shown in [3] that the fuzzy solution this equation is periodic for any choice of initial condition . According to the previous proposition, projections of fuzzy on are obtained by taking extensions of components Zadeh and .
Figure 3 shows the time evolution of the fuzzy projection of on and , respectively. Take the initial condition defined by the membership function.

Example 7. Consider the epidemiological model defined by equations
The solution of the model , defined by functions converges to the equilibrium point . According to what is discussed in [3], for all , the fuzzy solution converges to the equilibrium point fuzzy .
According to the equality (46), projections of the equilibrium point on the coordinate axis are obtained by extension of Zadeh components . That is, the projections are fuzzy, respectively, and , whose membership function is given by
By Proposition 5, fuzzy projections of fuzzy solution , on , of model are obtained by extension of Zadeh, the components and , given by
To illustrate, suppose the force infection is , and we take the initial condition defined by membership function
Figure 4 shows the evolution of applications and with the time evolution. Note that converges to , whereas converges to 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 , and, therefore, the fuzzy solution converge to the equilibrium point fuzzy . In this case, the projections and converges to and , respectively.
In Figure 5, we plot the projections of the fuzzy solution , to the initial condition and given by fuzzy set

The graphical representation of fuzzy projections of this work is established as follows: given an , the region in plan bounded by -level is filled with a shade of gray. If , then the region bounded by is filled with the white color, whereas if , then the region bounded by is filled with black. Thus, the larger the degree of membership of a point , 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 define the equation, and thus, the parameter vector now is a part of the initial condition. Thus, Zadeh’s extension to the flow generated by (57) incorporates the uncertainties of initial conditions and parameters of (56).

Once the solution generated by (56) is continuous in the initial condition and parameters, Zadeh's extension of is well defined, and according to (5), for all we have:

From the standpoint of applications, it is important to know the flow behavior of the deterministic phase space of (56) instead of space to (57), since the flow components , that are , do not have any additional information. It is worth noting that, for all , we have

Analogously to the deterministic case, we can also be interested only in the fuzzy flow behavior on the phase space . The fuzzy projections defined at the outset of this work can then be used to obtain the fuzzy flow behavior on the space .

The following statement characterizes the relationship between the projection of fuzzy on the space and Zadeh's extension: is a solution of (56).

Proposition 8. The application , given by Zadeh's extension , is the fuzzy projection of fuzzy flow on .

Proof. Let and fix . To prove the claim, it suffices to show that is the fuzzy projection of of , then .
To simplify, let be the image set of . By definition, the membership function of is given by
Let be the projection of on . By definition of fuzzy projection, the membership function of is given by Now, as , so, if and only if for some . So, for all , the membership function of is If , so, for all , and so, .
But the fuzzy projection of on has the membership function
Now, by definition, the point has the membership function
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 for all and . Using the continuity of applications and , we have for all . 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 , the -levels to the fuzzy flow generated by (56) are the sets On the other hand, if the -levels of 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 , in terms of and , is given by and thus the flow 2-dimensional , for the case in which the parameter is incorporated into the initial condition, is given by According to Proposition 8, Zadeh's extension of is the projection of fuzzy flow . To illustrate, consider . By definition, we have Moreover, the projection has -levels given by from which we conclude that , and consequently,
For any initial condition , we show that converges to the equilibrium points which is Zadeh's extension given by . That is, the equilibrium point has membership function In particular, if is the fuzzy cartesian product of and , then the membership function in this case is given by when and , when .
The projection for this equilibrium point has membership function and we have as .
In Figure 6, we have the graphical representation of the fuzzy solution and its fuzzy projection .

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.