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.
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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
in terms of the stability of the trivial solution of the ordinary difference equation
where 
is continuous in 
for each 
, and 
for each 
, where 
such that 
satisfies the following:
-
(i)
is normal; -
(ii)
is fuzzy convex; -
(iii)
is upper semicontinuous; -
(iv)
is compact, and 
is a continuous and nondecreasing function in 
for each 
.
is normal;
is fuzzy convex;
is upper semicontinuous;
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
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
of the deterministic system
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
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.
In [11] we have, already, investigated the behavior of the solutions of a related system of two parametric ordinary difference equations, of the form
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
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 
.
We denote by 
the set of functions 
such that,
where 
satisfies the following conditions:
-
(i)
is normal, that is, there exists an 
such that 
; -
(ii)
is fuzzy convex, that is for 
; 
(2.2) -
(iii)
is upper semicontinuous -
(iv)
The support of
, 
is compact.
is normal, that is, there exists an 
such that 
;
is fuzzy convex, that is for 
; 
is upper semicontinuous
, 
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 
,
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
-
(i)
is nondecreasing and left continuous; -
(ii)
is nonincreasing and left continuous; -
(iii)
.
is nondecreasing and left continuous;
is nonincreasing and left continuous;
.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:
-
(i)
, 
positive real numbers, -
(ii)
, 
positive real numbers, -
(iii)
, 
positive real numbers.
, 
positive real numbers,
, 
positive real numbers,
, 
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
-
(i)
is a positive fuzzy number which belongs to 
, with 
(2.5) -
(ii)
is a positive fuzzy number which belongs to 
, with 
(2.6)if
; -
(iii)
is a positive fuzzy number which belongs to 
, with 
(2.7)
is a positive fuzzy number which belongs to 
, with 
is a positive fuzzy number which belongs to 
, with 
;
is a positive fuzzy number which belongs to 
, with 
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
Let 
be positive fuzzy numbers which belong to 
such that (2.4) holds. We consider the following metric (see [29, 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).
We say that a positive fuzzy number 
is a positive equilibrium for (1.6) if
Let 
be a sequence of positive fuzzy numbers and 
is a positive fuzzy number. Suppose that
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
where
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
where
. We define the function 
such that
If 
we have that there exists a measurable set 
of measure zero such that
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
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.
-
(i)
Suppose that
(3.1)
(3.2)
(3.3)hold. Then
, 
. -
(ii)
Suppose that
(3.4)
(3.5)hold. Then
(3.6)
, 
. 
Proposition B (see [11]).
Consider the system of algebraic equations
Then the following statements are true.
-
(i)
If (3.2) holds, the system (3.7) has a unique nonnegative solution
. -
(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.
-
(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 
. -
(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 
.
of system (1.7) we have that 
tends to the unique zero equilibrium 
of (1.7) as 
.
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
Let 
be fuzzy numbers and 
, 
, 
positive real numbers such that
Then the following statements are true.
-
(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 
. -
(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 
.
of the fuzzy difference equation (1.6) with initial values 
.
of the fuzzy difference equation (1.6) with initial values 
.Proof.
We consider the family of systems of parametric ordinary difference equations for 
and 
,
-
(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
with initial values 
, 
, has a positive solution 
and so
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
and so from (3.10)–(3.13), and (3.17)
Moreover, from (3.10)–(3.13), (3.17), and (3.19), we get
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
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 
Finally, using (3.10), (3.11), (3.13), (3.17), (3.19), (3.20), (3.23), and working inductively, we get for 
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
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 
In addition, from (3.10), (3.23), and (3.26), we get
and so from (3.17), (3.23), and (3.26)
Therefore, using (3.28) and arithmetic multiplication on closed intervals
Using arithmetic operations on positive fuzzy numbers and (2.8) we have
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
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).
-
(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)
, 
has a positive solution 
such that (3.20) and 
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
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.
-
(i)
If
(3.36)then zero is the unique nonnegative equilibrium of the fuzzy difference equation (1.6).
-
(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.
where
hold.Proof.
We consider the fuzzy equation
where 
is a positive fuzzy number such that (3.10) holds. Suppose that 
, is a solution of (3.40) such that
Then using arithmetic operations on fuzzy numbers and (2.8), (3.10), we can easily prove that 
satisfies the family of parametric algebraic systems
-
(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).
-
(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).
, 
, 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
We consider the function
then
We can easily prove that
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
and so from (3.46)
which means that
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,
is a left continues function with respect to 
, 
.
Finally, from (3.39) we have that
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.
-
(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
. -
(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).
.
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
and since
where 
and 
is taken for all 
, from (3.55) and (3.57), we get
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
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 
where 
is a fuzzy number such that
We take the initial values 
such that
From (3.62), we get
and so
Moreover from (3.63) we take
and so
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
We take the initial values 
such that
From (3.68), we get
and so
Moreover from (3.69) we take
and so
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
Then from (3.74), we get
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
Then from (3.76), we get
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]).
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The authors would like to thank the referees for their helpful suggestions.
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Stefanidou, G., Papaschinopoulos, G. & Schinas, C. On an Exponential-Type Fuzzy Difference Equation. Adv Differ Equ 2010, 196920 (2010). https://doi.org/10.1155/2010/196920
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DOI: https://doi.org/10.1155/2010/196920