- Research Article
- Open Access

# 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

Our goal is to investigate the existence of the positive solutions, the existence of a nonnegative equilibrium, and the convergence of a positive solution to a nonnegative equilibrium of the fuzzy difference equation , , , where and the initial values belong in a class of fuzzy numbers.

## Keywords

- Difference Equation
- Fuzzy Number
- Positive Real Number
- Closed Interval
- Arithmetic Operation

## 1. Introduction

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

- (i)
is normal;

- (ii)
is fuzzy convex;

- (iii)
is upper semicontinuous;

- (iv)
is compact, and is a continuous and nondecreasing function in for each .

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 .

- (i)
is normal, that is, there exists an such that ;

- (ii)
- (iii)
is upper semicontinuous

- (iv)
The support of , is compact.

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]).

- (i)
is nondecreasing and left continuous;

- (ii)
is nonincreasing and left continuous;

- (iii)
.

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

- (i)
, positive real numbers,

- (ii)
, positive real numbers,

- (iii)
, positive real numbers.

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

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

where is taken for all .

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)
If (3.2) holds, the system (3.7) has a unique nonnegative solution .

- (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.

- (i)
hold. Then there exists a unique positive solution of the fuzzy difference equation (1.6) with initial values .

- (ii)
hold. Then there exists a unique positive solution of the fuzzy difference equation (1.6) with initial values .

Proof.

- (i)

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.

- (i)
- (ii)
are the only nonnegative equilibriums of the fuzzy difference equation (1.6), such that (2.11) and hold.

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
where , , is the unique function which satisfies (3.39).

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

## References

- Lakshmikantham V, Vatsala AS:
**Basic theory of fuzzy difference equations.***Journal of Difference Equations and Applications*2002,**8**(11):957-968. 10.1080/1023619021000048850MATHMathSciNetView ArticleGoogle Scholar - Buckley JJ, Feuring T, Hayashi Y:
**Fuzzy difference equations: the initial value problem.***Journal of Advanced Computational Intelligence*2001,**5**(6):315-325.Google Scholar - Bassanezi RC, de Barros LC, Tonelli PA:
**Attractors and asymptotic stability for fuzzy dynamical systems.***Fuzzy Sets and Systems*2000,**113**(3):473-483. 10.1016/S0165-0114(98)00142-0MATHMathSciNetView ArticleGoogle Scholar - Kocić VL, Ladas G:
*Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Mathematics and Its Applications*.*Volume 256*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:xii+228.Google Scholar - Beddington JR, Free CA, Lawton JH:
**Dynamic complexity in predator prey models framed in difference equations.***Nature*1975,**255**(5503):58-60. 10.1038/255058a0View ArticleGoogle Scholar - Cooke KL, Calef DF, Level EV:
**Stability or chaos in discrete epidemic models.**In*Nonlinear Systems and Applications*. Academic Press, New York, NY, USA; 1977:73-93.Google Scholar - Gu E-G:
**The nonlinear analysis on a discrete host-parasitoid model with pesticidal interference.***Communications in Nonlinear Science and Numerical Simulation*2009,**14**(6):2720-2727. 10.1016/j.cnsns.2008.08.012MATHMathSciNetView ArticleGoogle Scholar - Kendall BE:
**Cycles, chaos, and noise in predator-prey dynamics.***Chaos, Solitons and Fractals*2001,**12**(2):321-332. 10.1016/S0960-0779(00)00180-6MATHMathSciNetView ArticleGoogle Scholar - Zhang DC, Shi B:
**Oscillation and global asymptotic stability in a discrete epidemic model.***Journal of Mathematical Analysis and Applications*2003,**278**(1):194-202. 10.1016/S0022-247X(02)00717-5MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On a discrete epidemic model.***Discrete Dynamics in Nature and Society*2007,**2007:**-10.Google Scholar - Stefanidou G, Papaschinopoulos G, Schinas CJ:
**On a system of two exponential type difference equations.***Communications on Applied Nonlinear Analysis*2010,**17**(2):1-13.MATHMathSciNetGoogle Scholar - Stefanidou G, Papaschinopoulos G:
**The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation.***Information Sciences*2006,**176**(24):3694-3710. 10.1016/j.ins.2006.02.006MATHMathSciNetView ArticleGoogle Scholar - Berg L:
**On the asymptotics of nonlinear difference equations.***Zeitschrift für Analysis und ihre Anwendungen*2002,**21**(4):1061-1074.MATHView ArticleGoogle Scholar - Berg L:
**Inclusion theorems for non-linear difference equations with applications.***Journal of Difference Equations and Applications*2004,**10**(4):399-408. 10.1080/10236190310001625280MATHMathSciNetView ArticleGoogle Scholar - Berg L:
**On the asymptotics of the difference equation**.*Journal of Difference Equations and Applications*2008,**14**(1):105-108. 10.1080/10236190701503041MATHMathSciNetView ArticleGoogle Scholar - Gutnik L, Stević S:
**On the behaviour of the solutions of a second-order difference equation.***Discrete Dynamics in Nature and Society*2007,**2007:**-14.Google Scholar - Iričanin B, Stević S:
**Eventually constant solutions of a rational difference equation.***Applied Mathematics and Computation*2009,**215**(2):854-856. 10.1016/j.amc.2009.05.044MATHMathSciNetView ArticleGoogle Scholar - Kent CM:
**Convergence of solutions in a nonhyperbolic case.***Nonlinear Analysis: Theory, Methods & Applications*2001,**47**(7):4651-4665. 10.1016/S0362-546X(01)00578-8MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On the recursive sequence**.*Taiwanese Journal of Mathematics*2002,**6**(3):405-414.MATHMathSciNetGoogle Scholar - Stević S:
**Global stability and asymptotics of some classes of rational difference equations.***Journal of Mathematical Analysis and Applications*2006,**316**(1):60-68. 10.1016/j.jmaa.2005.04.077MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**On monotone solutions of some classes of difference equations.***Discrete Dynamics in Nature and Society*2006,**2006:**-9.Google Scholar - Stević S:
**On positive solutions of a****-th order difference equation.***Applied Mathematics Letters*2006,**19**(5):427-431. 10.1016/j.aml.2005.05.014MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Asymptotics of some classes of higher-order difference equations.***Discrete Dynamics in Nature and Society*2007,**2007:**-20.Google Scholar - Stević S:
**Existence of nontrivial solutions of a rational difference equation.***Applied Mathematics Letters*2007,**20**(1):28-31. 10.1016/j.aml.2006.03.002MATHMathSciNetView ArticleGoogle Scholar - Stević S:
**Nontrivial solutions of higher-order rational difference equations.***Matematicheskie Zametki*2008,**84**(5):772-780.MathSciNetView ArticleGoogle Scholar - Sun T:
**On non-oscillatory solutions of the recursive sequence****.***Journal of Difference Equations and Applications*2005,**11**(6):483-485. 10.1080/10236190512331319343MATHMathSciNetView ArticleGoogle Scholar - Voulov HD:
**Existence of monotone solutions of some difference equations with unstable equilibrium.***Journal of Mathematical Analysis and Applications*2002,**272**(2):555-564. 10.1016/S0022-247X(02)00173-7MATHMathSciNetView ArticleGoogle Scholar - Nguyen HT, Walker EA:
*A First Course in Fuzzy Logic*. CRC Press, Boca Raton, Fla, USA; 1997:viii+266.MATHGoogle Scholar - Wu C, Zhang B:
**Embedding problem of noncompact fuzzy number space****. I.***Fuzzy Sets and Systems*1999,**105**(1):165-169. 10.1016/S0165-0114(97)00218-2MATHMathSciNetView ArticleGoogle Scholar - Klir GJ, Yuan B:
*Fuzzy Sets and Fuzzy Logic: Theory and Applications*. Prentice Hall, Upper Saddle River, NJ, USA; 1995:xvi+574.MATHGoogle Scholar - Bede B, Tanali GB:
*Perspectives of Fuzzy Initial Value Problems, Research Reports Series*. University of Texas at El Paso; 2006.Google Scholar - Diamond P, Kloeden P:
*Metric Spaces of Fuzzy Sets: Theory and Application*. World Scientific, River Edge, NJ, USA; 1994:x+178.View ArticleGoogle Scholar - Román-Flores H, Barros LC, Bassanezi RC:
**A note on Zadeh's extensions.***Fuzzy Sets and Systems*2001,**117**(3):327-331. 10.1016/S0165-0114(98)00408-4MATHMathSciNetView ArticleGoogle Scholar - Papaschinopoulos G, Stefanidou G:
**Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation.***Fuzzy Sets and Systems*2003,**140**(3):523-539. 10.1016/S0165-0114(03)00034-4MATHMathSciNetView ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.