# On an Exponential-Type Fuzzy Difference Equation

- G Stefanidou
^{1}, - G Papaschinopoulos
^{1}Email author and - CJ Schinas
^{1}

**2010**:196920

https://doi.org/10.1155/2010/196920

© G. Stefanidou et al. 2010

**Received: **11 March 2010

**Accepted: **24 October 2010

**Published: **27 October 2010

## Abstract

## Keywords

## 1. Introduction

Fuzzy difference equations are approached by many authors, from a different view.

- (i)
- (ii)
- (iii)
- (iv)

for , where is the unknown function of and are real constants with . is a known function of and parameters , which is continuous in . The initial conditions are fuzzy sets.

where is the Zadeh's extensions of a continuous function . Equations (1.4) and (1.5) have the same real constants coefficient and real equilibriums.

where and the initial values are in a class of fuzzy numbers (see Preliminaries). This equation is motivated by the corresponding ordinary difference equation which is posed in [4]. Moreover, (1.6) is a special case of an epidemic model (see [5–8]) and was studied in [9] by Zhang and Shi and in [10] by Stević.

where are positive real numbers and the initial values , , , are positive real numbers, which satisfy some additional conditions.

where are positive integers, , and the initial values , , are in a class of fuzzy numbers, under some conditions has unbounded solutions, something that does not happen in the case of the corresponding ordinary difference equation (1.8), where are positive integers and , and the initial values , , are positive real numbers.

Finally we note that in recent years there has been a considerable interest in the study of the existence of some specific classes of solutions of difference equations such as nontrivial, nonoscillatory, monotone, positive. Various methods have been developed by the experts. For partial review of the theory of difference equations and their applications see, for example, [4, 10, 13–27] and the references therein.

## 2. Preliminaries

For a set , we denote by the closure of .

are closed intervals. Obviously, .

We say that a fuzzy number is positive if .

To prove our main results, we need the following theorem (see [29]).

Theorem 2.1 (see [29]).

Conversely, for any functions defined in which satisfy (i)–(iii) in above and is compact, there exists a unique such that , .

- (i)
- (ii)
- (iii)

- (i)
- (ii)
- (iii)

We note that the subtraction "−" we use, is different than Hukuhara difference (see [31, 32]).

We say is a positive solution of (1.6) if is a sequence of positive fuzzy numbers which satisfies (1.6).

If , we say that converges to with respect to as .

We consider however, two fuzzy numbers to be equivalent if there exists a measurable set of measure zero such that (2.16) hold and if we do not distinguish between equivalent of fuzzy numbers then becomes a metric space with metric .

## 3. Study of the Fuzzy Difference Equation (1.6)

In order to prove our main results, we need the following Propositions A, B, C, which can be found in [11]. For readers convenience, we cite them below without their proofs.

Proposition A (see [11]).

- (i)
- (ii)

Proposition B (see [11]).

- (i)
- (ii)

hold; then there are only two nonnegative equilibriums of system (3.7), such that , which are , , .

Proposition C (see [11]).

- (i)If (3.2) and either (3.1) and (3.3) or (3.5) are satisfied, then for the solution of system (1.7) we have that
holds and obviously tends to the unique zero equilibrium of (1.7) as .

- (ii)
Suppose that (3.5), the first relation of (3.2) and the second relation of (3.8) are satisfied. Then tends to the nonnegative equilibrium , of (1.7) as .

First we study the existence and the uniqueness of the positive solutions of the fuzzy difference equation (1.6).

Proposition 3.1.

Proof.

In addition, from (3.10)–(3.14) and Proposition A, we have that (3.17) has a positive solution , , with initial values , . We prove that , determines a sequence of positive fuzzy numbers.

Since , and are positive fuzzy numbers, from Theorem 2.1 we have that , , and , , are left continues and so from (3.17), we get that are left continuous as well.

where is the solution of (3.19).

and thus, our claim is true.

- (ii)

hold.

We prove that , determines a sequence of positive fuzzy numbers.

From (3.10), (3.11), (3.15)–(3.17), (3.19), and (3.20), we get that (3.23) holds. Moreover, arguing as in statement (i), we can easily prove that determine a positive fuzzy number such that (3.24) holds.

As in statement (i), using (3.10), (3.11), (3.15)–(3.17), (3.24), Theorem 2.1 and working inductively, we get that the positive solution , , , of (3.17), determines a sequence of positive fuzzy numbers , such that (3.27) holds.

Finally, arguing as in statement (i) we have that is the unique positive solution of the fuzzy difference equation (1.6) with initial values , , such that (3.10), (3.11), (3.15) and (3.16) hold. This completes the proof of the proposition.

In the next proposition we study the existence of nonnegative equilibriums of the fuzzy difference equation (1.6).

Proposition 3.2.

Proof.

- (i)If (3.36) holds then from (3.10), (3.41), (3.42), and statement (i) of Proposition B, we get that
This completes the proof of statement (i).

- (ii)If (3.37) and (3.41) hold then from (3.10) and statement (ii) of Proposition B, we get that system (3.42) has only two solutions, which are

Using (3.41) and (3.44) we have that zero is a solution of the fuzzy equation (3.40).

since is an increasing function. From (3.52) it is obvious that is a decreasing function with respect to , .

In addition, since is a continuous and increasing function, we have that is also a continuous and increasing function. Moreover, is a left continuous function with respect to , .

is a left continues function with respect to , .

From Theorem 2.1, (3.39), (3.52), (3.54), and since is a left continuous function with respect to , , we have that , determines a fuzzy number such that (3.38) holds. Therefore, from (3.45) is a solution of the fuzzy equation (3.40). This completes the proof of the proposition.

In the last proposition we study the asymptotic behavior of the positive solutions of the fuzzy difference equation (1.6).

Proposition 3.3.

- (i)
and either (3.12) and (3.14) or (3.16) are satisfied. Then every positive solution of the fuzzy difference equation (1.6) tends to the zero equilibrium as .

- (ii)
and (3.16) are satisfied. Then every positive solution of the fuzzy difference equation (1.6) nearly converges to the nonnegative equilibrium with respect to as and converges to with respect to as , where was defined by (3.38) and (3.39).

Proof.

(i) Since (3.55) and either (3.12) and (3.14) or (3.16) are satisfied, from Proposition 3.1 the fuzzy difference equation (1.6) has unique positive solution , such that (3.27) holds.

In addition, (3.10) and (3.55) imply that (3.36) holds. So, from statement (i) of Proposition 3.2, zero is the unique nonnegative equilibrium of the fuzzy difference equation (1.6).

This completes the proof of statement (i).

Using (3.60) and arguing as in Proposition of [34], we can prove that the positive solution of (1.6) nearly converges to with respect to as and converges to with respect to as . Thus, the proof of the proposition is completed.

To illustrate our results we give some examples in which the conditions of our propositions hold.

Example 3.4.

Therefore the conditions (3.10)–(3.14) are satisfied. So from statement (i) of Proposition 3.1 the solution of (3.61) with initial values is positive and unique. In addition it is obvious that (3.36) are satisfied. Then from the statement (i) of Proposition 3.2 we have that zero is the unique nonnegative equilibrium of (3.61). Finally from Proposition 3.3 the unique positive solution of (3.61) with initial values tends to the zero equilibrium of (3.61) as .

Example 3.5.

Therefore the conditions (3.15), (3.16) are satisfied. So from statement (ii) of Proposition 3.1 the solution of (3.61) with initial values is positive and unique.

Example 3.6.

Then it is obvious that (3.37) are satisfied. Then from the statement (ii) of Proposition 3.2 we have that zero and where , , , are the only nonnegative equilibriums of the fuzzy difference equation (3.61), such that (2.11) and hold.

Example 3.7.

We consider the fuzzy difference equation (3.61) where is given by (3.62). Let , be the fuzzy numbers given by (3.69). Then since (3.15), (3.16), and (3.36) hold from Propositions 3.1, 3.2 and 3.3 the unique positive solution of (3.61) with initial values tends to the zero equilibrium of (3.61) as .

Example 3.8.

Then it is obvious that relations (3.37) are satisfied. So from the statement (ii) of Proposition 3.2 we have that zero and where , , , are the only nonnegative equilibriums of the fuzzy difference equation (3.61), such that (2.11) and hold. Let be the fuzzy numbers defined in (3.69). Then from the statement (ii) of Proposition 3.1 and statement (ii) of Proposition 3.3 we have that the unique positive solution of (3.61) with initial values nearly converges to the nonnegative equilibrium with respect to as and converges to with respect to as .

## 4. Conclusions

In this paper, we considered the fuzzy difference equation (1.6), where and the initial values are positive fuzzy numbers. The corresponding ordinary difference equation (1.6) is a special case of an epidemic model. The combine of difference equations and Fuzzy Logic is an extra motivation for studying this equation. A mathematical modelling of a real world phenomenon, very often, leads to a difference equation and on the other hand, Fuzzy Logic can handle uncertainness, imprecision or vagueness related to the experimental input-output data.

The main results of this paper are the following. Firstly, under some conditions on and initial values we found positive solutions and nonnegative equilibriums and then we studied the convergence of the positive solutions to the nonnegative equilibrium of the fuzzy difference equations (1.6). We note that, in order to study the fuzzy difference equation (1.6), we used the results concerning the behavior of the solutions of the related system of two parametric ordinary difference equations (1.7) (see [11]).

## Declarations

### Acknowledgment

The authors would like to thank the referees for their helpful suggestions.

## Authors’ Affiliations

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