Open Access

On an Exponential-Type Fuzzy Difference Equation

Advances in Difference Equations20102010:196920

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

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.

1. Introduction

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

In [1], the authors developed the stability results for the fuzzy difference equation
(1.1)
in terms of the stability of the trivial solution of the ordinary difference equation
(1.2)
where is continuous in for each , and for each , where such that satisfies the following:
  1. (i)

    is normal;

     
  2. (ii)

    is fuzzy convex;

     
  3. (iii)

    is upper semicontinuous;

     
  4. (iv)

    is compact, and is a continuous and nondecreasing function in for each .

     
In [2], the authors studied the second-order, linear, constant coefficient fuzzy difference equation of the form
(1.3)

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.

In [3] the authors considered the associated fuzzy system
(1.4)
of the deterministic system
(1.5)

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.

In this paper, we consider the fuzzy difference equation
(1.6)

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 [58]) and was studied in [9] by Zhang and Shi and in [10] by Stević.

In [11] we have, already, investigated the behavior of the solutions of a related system of two parametric ordinary difference equations, of the form
(1.7)

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

We note that, the behavior of the fuzzy difference equation is not always the same with the corresponding ordinary difference equation. For instance, in paper [12] the fuzzy difference equation
(1.8)

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, 1327] and the references therein.

2. Preliminaries

For a set , we denote by the closure of .

We denote by the set of functions such that,
(2.1)
where satisfies the following conditions:
  1. (i)

    is normal, that is, there exists an such that ;

     
  2. (ii)
    is fuzzy convex, that is for ;
    (2.2)
     
  3. (iii)

    is upper semicontinuous

     
  4. (iv)

    The support of , is compact.

     
Obviously, set is a class of fuzzy numbers. In this paper, all the fuzzy numbers we use are elements of . From above (i)–(iv) and Theorems and of [28] the -cuts of the fuzzy number ,
(2.3)

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]).

Let , such that , . Then can be regarded as functions on which satisfy
  1. (i)

    is nondecreasing and left continuous;

     
  2. (ii)

    is nonincreasing and left continuous;

     
  3. (iii)

    .

     

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

We need the following arithmetic operations on closed intervals:
  1. (i)

    , positive real numbers,

     
  2. (ii)

    , positive real numbers,

     
  3. (iii)

    , positive real numbers.

     
In this paper, we use the following arithmetic operations on fuzzy numbers based on closed intervals arithmetic (see [30]). Let be positive fuzzy numbers which belong to with
(2.4)
  1. (i)
    is a positive fuzzy number which belongs to , with
    (2.5)
     
  2. (ii)
    is a positive fuzzy number which belongs to , with
    (2.6)

    if ;

     
  3. (iii)
    is a positive fuzzy number which belongs to , with
    (2.7)
     

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

Using Extension Principle (see [28, 30, 33]) for a positive fuzzy number such that (2.4) holds, we have
(2.8)
Let be positive fuzzy numbers which belong to such that (2.4) holds. We consider the following metric (see [29, 32]):
(2.9)

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).

We say that a positive fuzzy number is a positive equilibrium for (1.6) if
(2.10)
Let be a sequence of positive fuzzy numbers and is a positive fuzzy number. Suppose that
(2.11)
are satisfied. We say that nearly converges to with respect to as if for every there exists a measurable set , of measure less than such that
(2.12)
where
(2.13)

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

Let be the set of positive fuzzy numbers. From Theorem 2.1 we have that (resp., ) are increasing (resp., decreasing) functions on . Therefore, using the condition (iv) of the definition of the fuzzy numbers there exist the Lebesque integrals
(2.14)
where . We define the function such that
(2.15)
If we have that there exists a measurable set of measure zero such that
(2.16)

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 .

We say that a sequence of positive fuzzy numbers converges to a positive fuzzy number with respect to as if
(2.17)

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]).

Consider system (1.7) where the constants are positive real numbers. Let be a solution of (1.7) with initial values , , , . Then the following statements are true.
  1. (i)
    Suppose that
    (3.1)
    (3.2)
    (3.3)

    hold. Then , .

     
  2. (ii)
    Suppose that
    (3.4)
    (3.5)
    hold. Then
    (3.6)
     

Proposition B (see [11]).

Consider the system of algebraic equations
(3.7)
Then the following statements are true.
  1. (i)

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

     
  2. (ii)
    Suppose that
    (3.8)
     

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

Proposition C (see [11]).

Consider system (1.7). Let be a solution of (1.7). Then the following statements are true.
  1. (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
    (3.9)

    holds and obviously tends to the unique zero equilibrium of (1.7) as .

     
  2. (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.

Consider the fuzzy difference equation (1.6), where is a positive fuzzy number such that
(3.10)
Let be fuzzy numbers and , , positive real numbers such that
(3.11)
Then the following statements are true.
  1. (i)
    Suppose that
    (3.12)
    (3.13)
    (3.14)

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

     
  2. (ii)
    Suppose that
    (3.15)
    (3.16)

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

     

Proof.

We consider the family of systems of parametric ordinary difference equations for and ,
(3.17)
  1. (i)
    From (3.11) and (3.14), we can consider that
    (3.18)
     
Using relations (3.10)–(3.13), (3.18), and Proposition A, we get that the system of ordinary difference equations
(3.19)
with initial values , , has a positive solution and so
(3.20)

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.

In addition, for any , , we have
(3.21)
and so from (3.10)–(3.13), and (3.17)
(3.22)
Moreover, from (3.10)–(3.13), (3.17), and (3.19), we get
(3.23)
Therefore, from Theorem 2.1 relations (3.22), (3.23), and since are left continuous, we have that determine a positive fuzzy number such that
(3.24)
Since , are left continuous from (3.17) and working inductively, we get that , , are also left continuous. In addition, using (3.10), (3.11), (3.13), (3.17), (3.20), (3.21), (3.22), and working inductively, we get for any , and
(3.25)
Finally, using (3.10), (3.11), (3.13), (3.17), (3.19), (3.20), (3.23), and working inductively, we get for
(3.26)

where is the solution of (3.19).

Therefore, since , , are left continuous and (3.22), (3.23), (3.25), (3.26) are satisfied, from Theorem 2.1, we get that the positive solution , , , of (3.17), with initial values , , , satisfying (3.11), (3.12), (3.14), determines a sequence of positive fuzzy numbers , such that
(3.27)
We claim that is a solution of (1.6) with initial values , , such that (3.11), (3.12), and (3.14) hold. From (3.17) and (3.27) we have for all
(3.28)
In addition, from (3.10), (3.23), and (3.26), we get
(3.29)
and so from (3.17), (3.23), and (3.26)
(3.30)
Therefore, using (3.28) and arithmetic multiplication on closed intervals
(3.31)
Using arithmetic operations on positive fuzzy numbers and (2.8) we have
(3.32)

and thus, our claim is true.

Finally, suppose that there exists another solution of the fuzzy difference equation (1.6) with initial values , , such that (3.10)–(3.14) hold. Then using the uniqueness of the solutions of the system (3.17) and arithmetic operations on positive fuzzy numbers and (2.8), we can easily prove that
(3.33)
and so we have that is the unique positive solution of the fuzzy difference equation (1.6) with initial values , , such that (3.11), (3.12), and (3.14) hold. This completes the proof of statement (i).
  1. (ii)
    From (3.10), (3.15), (3.16), and Proposition A, we get that system (3.19) with initial values , has a positive solution such that (3.20) and
    (3.34)
     

hold.

From (3.10), (3.11), (3.15), (3.16), and Proposition A, we have that (3.17) has a positive solution , , with initial values , , such that
(3.35)

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.

Consider the fuzzy difference equation (1.6) where is a positive fuzzy number such that (3.10) holds and the initial values , are positive fuzzy numbers. Then the following statements are true.
  1. (i)
    If
    (3.36)

    then zero is the unique nonnegative equilibrium of the fuzzy difference equation (1.6).

     
  2. (ii)
    If
    (3.37)
    then zero and where
    (3.38)
    (3.39)

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

     

Proof.

We consider the fuzzy equation
(3.40)
where is a positive fuzzy number such that (3.10) holds. Suppose that , is a solution of (3.40) such that
(3.41)
Then using arithmetic operations on fuzzy numbers and (2.8), (3.10), we can easily prove that satisfies the family of parametric algebraic systems
(3.42)
  1. (i)
    If (3.36) holds then from (3.10), (3.41), (3.42), and statement (i) of Proposition B, we get that
    (3.43)

    This completes the proof of statement (i).

     
  2. (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
    (3.44)
    (3.45)

    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).

To continue, we have to prove that , , determines a fuzzy number, where , satisfies (3.39). From (3.39), we get
(3.46)
We consider the function
(3.47)
then
(3.48)
We can easily prove that
(3.49)
is an increasing and positive function for and so using (3.48), we get that is an increasing function for . Since is a positive, decreasing function with respect to , , we get that
(3.50)
and so from (3.46)
(3.51)
which means that
(3.52)

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 , .

Therefore,
(3.53)

is a left continues function with respect to , .

Finally, from (3.39) we have that
(3.54)

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.

Consider the fuzzy difference equation (1.6) where is a positive fuzzy number such that (3.10) holds. Let , be the initial values such that (3.11) holds. Then the following statements are true.
  1. (i)
    Suppose that
    (3.55)

    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 .

     
  2. (ii)
    Suppose that
    (3.56)

    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).

From the analogous relation of (3.9) of Proposition C and using (3.10), (3.11), we get
(3.57)
and since
(3.58)
where and is taken for all , from (3.55) and (3.57), we get
(3.59)

This completes the proof of statement (i).

(ii) Since from (3.56), we have that (3.15) and (3.37) are fulfilled, we get from (3.16) and statement (ii) of Propositions 3.1 and 3.2 that the fuzzy difference equation (1.6) has unique positive solution such that (3.27) holds, and a nonnegative equilibrium , such that (3.38) and (3.39) hold. Since is a positive solution of system (3.17), from (3.11), (3.16), (3.56), and Proposition C we have that
(3.60)

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.

Consider the fuzzy equation (1.6) for
(3.61)
where is a fuzzy number such that
(3.62)
We take the initial values such that
(3.63)
From (3.62), we get
(3.64)
and so
(3.65)
Moreover from (3.63) we take
(3.66)
and so
(3.67)

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.

Consider the fuzzy equation (3.61) where is a fuzzy number such that
(3.68)
We take the initial values such that
(3.69)
From (3.68), we get
(3.70)
and so
(3.71)
Moreover from (3.69) we take
(3.72)
and so
(3.73)

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.

We consider equation (3.61) where the fuzzy number is given as follows
(3.74)
Then from (3.74), we get
(3.75)

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.

Consider the fuzzy difference equation (3.61) where the fuzzy number is given by
(3.76)
Then from (3.76), we get
(3.77)

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

(1)
School of Engineering, Democritus University of Thrace

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© G. Stefanidou et al. 2010

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