ON A NONLINEAR FUZZY DIFFERENCE EQUATION

In this paper we investigate the existence, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation zn+1 = Azn−1 1 + z n−2 , n ∈ N0 where (zn) is a sequence of positive fuzzy numbers, A and the initial conditions z−j (j = 0, 1, 2) are positive fuzzy numbers and p is a positive integer.


Introduction
Over the last two decades, a lot of study has been published on difference equations and systems. One reason for this is that such equations and systems have high applicability both in mathematics and other sciences such as population biology, economics, probability theory, genetics, psychology etc., (see, e.g., [2,6,14,15] and the references therein). In this way, many real life problems are modeled by means of difference equations and systems. In some cases, however, measurements or data on a problem may reveal uncertainty or the problem considered may require subjective interpretations. In such cases, a fuzzy difference equation model can be established using notion of fuzzy set. In this way, the uncertainty is modeled.
Fuzzy set theory has recently become a popular subject due to the increasing number of applications in technology, mathematics and other sciences. The part that we are interested in is, of course, that the notion of fuzzy set can be easily applied to difference equations. With this application, a powerful method for determining the behavior of solutions of difference equations emerges. Some studies using the method will be summarized below.
In [3], Deeba et al. studied the fuzzy analog of a difference equation which arises in population genetics. More precisely they studied the first order difference equation x n+1 = wx n + q, n ∈ N 0 (1) where (x n ) is a sequence of fuzzy numbers and w, q, x 0 are fuzzy numbers. Also, they discussed the fuzzy nonlinear difference equation where (x n ) is a sequence of fuzzy numbers, w, q, x 0 are fuzzy numbers and f : a is the set of all real numbers greater or equal to a, is a continuous and nondecreasing function in its arguments.
In [4], Deeba and Korvin studied the second order difference equation where (x n ) is a sequence of fuzzy numbers and a, b, c, x 0 , x −1 are fuzzy numbers. This equation is a linearized model of a nonlinear model which determines the carbondioxide (CO 2 ) level in the blood.
In [12], Papaschinopoulos and Papadopoulos studied the existence, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation where (x n ) is a sequence of fuzzy numbers and A and the initial conditions x −j (j = 0, 1, . . . , m) are fuzzy numbers for m ∈ N 1 . For more works on fuzzy difference equations, see the references [9,11,13] and the references cited therein.
In [5], El-Owaidy et al. investigated the global behavior of the difference equation where the nonnegative parameters and nonnegative initial conditions. Moreover, in [7], Gümüş and Soykan investigated the behavior of solutions of the system of difference equations where the positive parameters α, β, γ, α 1 , β 1 , γ 1 , p and the initial conditions u −i , v −i for i = 0, 1, 2 are positive real numbers. Note that system (6) can be reduced to the following system of difference equations by the change of variables u n = (β 1 /γ 1 ) 1/p x n and v n = (β/γ) 1/p y n with r = α/β and s = α 1 /β 1 . So, in order to study system (6), they investigated system (7). In this paper we investigate the existence, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation where (z n ) is a sequence of positive fuzzy numbers, A and the initial conditions z −j (j = 0, 1, 2) are positive fuzzy numbers and p is a positive integer.

Preliminaries
In this section, we give some definitions which will be used in this paper. For more details see [1,8,10,16].
Let us denote by R F the space of all fuzzy numbers. For 0 < α ≤ 1 and A ∈ R F , we denote α-cuts of fuzzy number A by where sup is taken for all α ∈ (0, 1]. Then from the above we take the following metric (b) Let (x n ) be a sequence of positive fuzzy numbers and x is a fuzzy number. Then we say that lim The following lemma and definition are given in [8]: for α ∈ (0, 1] be the α-cuts of X, Y , respectively. Let Z be a fuzzy number such that (b) We say that (x n ) for n ∈ N 0 is an unbounded sequence if the ||x n || for n ∈ N 0 is an unbounded sequence.
We need the following lemma which has been proved in [12].

Main Results
In this section, we prove our main results. Firstly, we will study the existence of the positive solutions of equation (8). We say (z n ) is a positive solution of equation (8) if (z n ) is a sequence of positive fuzzy numbers which satisfies equation (8).
Theorem 1. Consider equation (8) where A is a positive fuzzy number. Then for any positive fuzzy numbers z −j (j = 0, 1, 2) there exists a unique positive solution (z n ) of (8) with the initial conditions z −j (j = 0, 1, 2) .
In the following lemma, we will study the boundedness and persistence of the positive solutions of system (7): 74İ. YALÇ INKAYA, V. Ç ALIŞKAN, D. T. TOLLU Lemma 3. Assume that r, s ∈ (0, 1), then every positive solution of system (7) is bounded and persists.

Numerical Examples
In this section, we give two numerical examples for the solutions of equation (8) regard to the different values of A for p = 1 with the inital conditions z −j (j = 0, 1, 2) are satisfied  (8) for p = 1 where z n is a sequence of positive fuzzy numbers, the initial conditions z −j (j = 0, 1, 2) are satisfied (28) and A is satisfied From (29), we get for all α ∈ (0, 1]. There exists a unique solution of equation (8) by Theorem 1. Since A r,α < 1 for all α ∈ [0, 1], then by Theorem 5, the positive solution (z n ) of equation (8)     Example 2. Consider equation (8) for p = 1 where z n is a sequence of positive fuzzy numbers, the initial conditions z −j (j = 0, 1, 2) are satisfied (28) and A is satisfied From (30), we get [A] α = [α + 2, 4 − α] for all α ∈ (0, 1]. There exists a unique positive solution of equation (8) by Theorem 1. It is easy to see that for all α ∈ (0, 1], we have A l,α > 1. So, by case (ii) in Theorem 3, equation (8)

Conclusion
In this study, we investigated behavior of the fuzzy difference equation z n+1 = Az n−1 /(1 + z p n−2 ), where (z n ) is a sequence of positive fuzzy numbers, A and the initial conditions z −j (j = 0, 1, 2) are positive fuzzy numbers and p is a positive integer. We have shown that, under certain conditions, the positive solutions of this equation converge to zero. Also, we have considered the case where the solutions are unbounded. Finally, we have supported our theoretical results.

Author Contribution Statements
The authors contributed equally to the paper. All authors read and approved the final version of the paper.

Declaration of Competing Interests
The authors declare that they have no competing interests.