Large time behavior of solution to second-order fractal difference equation with positive fuzzy parameters

Abstract

The article is concerned with large time behavior of solution to second-order fractal difference equation with positive fuzzy parameters $x_{n + 1} = \frac{A + x_{n}}{B + x_{n - 1}}, n = 1, 2, \dots,$ here the initial values x_i (i = -1, 0) and the parameters A, B are positive fuzzy numbers. Utilizing a generalization of division (g-division) of fuzzy numbers, one presents large time behaviors of positive fuzzy solution including persistence, boundedness, global convergence. Moreover, two numerical examples verify the effectiveness of the qualitative analysis.

Keywords

Second- order fractal difference equation g-division large time behavior positive fuzzy parameter

1 Introduction

Difference equations or discrete systems are one of the most important classical dynamical models in applied mathematics, which have wide applications including computer science, biology, economics, infectious disease, etc (see [18–21, 33]). In the past decades, many mathematicians are interested in describing large time behavior of difference equations such as the oscillatory behavior, periodicity, bifurcation, stability, chaos, and the boundedness. Moreover similar results on large time behaviors of solution have been obtained for many nonlinear difference equation system [7, 10, 22, 24, 35, 38, 39].

In real world, the parameters of model often stem from statistical data, based on the experimental data obtained and the statistical method adopted. These models, even in the classic way, often suffer from uncertainty (fuzzy uncertainty) that can be manifested in state variables, initial conditions and system parameters.

In fact, many experts have accepted to cope with uncertainty, and they also affirm the fact that uncertainty often influences large time behaviors of the solution of mathematical model. Contributing mathematical model of the real life problems in such method, it often involves intrinsically uncertainty or vagueness. Up to now, the fuzzy set instituted by Zadeh [16] has become a powerful tool to various theories and many applications including fuzzy difference equations (FDE) and fuzzy differential equations.

FDE is a special type difference equation that the initial values, state variable, or parameters of model are fuzzy numbers and its solutions are a sequence of fuzzy numbers. Since FDE can be used to analyze the phenomena with inherent uncertainty, the study on this model has become an important topic both in theory and in its applications. In recent years, more and more literatures have been published on large time behavior of FDE (see [1, 2, 4, 8, 9, 11–13, 15, 25–32, 34, 36, 37, 40]).

As far as FDE is concerned, let’s briefly review its development. In 1996, Deeba, Korvin, and Koh [8] presented a FDE and its application in economic. In 1999, Deeba and Korvin [9] investigated subsequently the levels of carbon dioxide in blood by FDE model. Papaschinopoulos and Papadopoulos [11, 12] studied first-order and high-order fractal FDE respectively. It is important to point out that Lakshmikantham and Vatsala [37] put forward the basic theory of FDE. Constructing a Lyapunov type function, they proved the comparison theorem and stability principle of FDE in terms of crisp difference equation. As we all know, a classical way to consider uncertainty is the fuzzification of difference equations. The operators such as addition, subtraction, multiplication, and division followed by Zadeh extension principle. There are many literatures on large time behavior of FDE in [15, 26–28, 34]. For example, using Zadeh extension principle, Zhang, Yang and Liao [28] were concerned with large time behaviors of first-order Riccati FDE.

However, some researchers argue that the operations are not invertible including the standard Minkowski addition and multiplication. To overcome these defects, Stefanini [17] proposed a generation of division named g-division and founded a new way to study FDE. Since then, utilizing g-division, many scholars [3, 29–32] studied some nonlinear FDE and obtained desirable results. Compared with using Zadeh extension principle, the merits of g-division is that it can decrease the singularity of fuzzy solution due to reduction of the length of the support interval.

Inspired with the previous works, utilizing g-division, in this article, we further explore large time behaviors of solutions to a second-order fractal FDE. $x_{n + 1} = \frac{A + x_{n}}{B + x_{n - 1}}, n = 0, 1, 2, \dots,$ (1) where the initial values x_i (i = -1, 0) and the parameters A, B are positive fuzzy numbers.

The outline of this paper is arranged as follows. Sect.2 introduces the basic concepts of fuzzy set used throughout this paper. Section 3 investigates the large time behaviors of solutions to the second-order fractal FDE by means of g-division of fuzzy numbers. Section 4 gives two numerical simulation examples using MATLAB to support our theoretical analysis. Section 5 draws a general conclusion and future work.

2 Preliminary

In the section, we first review some basic concepts used in the sequel, for more detail, readers can refer to [3, 8, 16].

A function defined as U : R → [0, 1] is said to be a fuzzy number if it is normal, convex, upper semicontinuous, and compactly support on R.

For α ∈ (0, 1], the α-level of fuzzy number U is denoted by [U] _α = {x ∈ R : U (x) ≥ α}, and for α = 0, the support of U is written as $supp U = [U]_{0} = \bar{{x \in R | U (x) > 0}}$ .

It is clear that the [U] _α = [U_l,α, U_r,α], is a bounded closed interval in R, where U_l,α (resp. U_r,α) denote the left-hand (resp. right-hand) endpoint of the α-level sets [U] _α . U is said to be a positive fuzzy number if suppU ⊂ (0, ∞) . Particularly, if U is a trivial positive fuzzy number (positive real number), then the α-level sets [U] _α = [U, U] , for all α ∈ (0, 1] .

Let U, V be fuzzy numbers with [U] _α = [U_l,α, U_r,α] , [V] _α = [V_l,α, V_r,α], the norm of fuzzy numbers space is defined by $∥ U ∥ = sup_{α \in [0, 1]} max {| U_{l, α} |, | U_{r, α} |} .$ The distance is $D (U, V) = sup_{α \in [0, 1]} max {| U_{l, α} - V_{l, α} |, | U_{r, α} - V_{r, α} |} .$ The addition and multiplication is defined by $[U + V]_{α} = [U_{l, α} + V_{l, α}, U_{r, α} + V_{r, α}],$ $[kU]_{α} = [{kU}_{l, α}, {kU}_{r, α}], k > 0$

All fuzzy numbers with addition and multiplication is represented by $R_{F}$ . The fuzzy number U is called positive (negative) if suppU ⊂ (0, + ∞) (suppU ⊂ (- ∞ , 0)). We denote by $R_{F}^{+} (R_{F}^{-})$ , the space of all positive (negative) fuzzy numbers.

The definition of g-division of fuzzy numbers is from [17].

Given $U, V \in R_{F}$ with [U] _α = [U_l,α, U_r,α] , [V] _α = [V_l,α, V_r,α], 0 ∉ [V] _α, ∀ α ∈ [0, 1]. The g-division (÷_g) is defined as a fuzzy number W = U ÷ _gV with [W] _α = [W_l,α, W_r,α](here [U] _α^-1 = [1/U_r,α, 1/U_l,α]), and $\begin{matrix} [W]_{α} = [U]_{α} \div_{g} [V]_{α} \Leftrightarrow \\ {\begin{matrix} (i) & [U]_{α} = [V]_{α} [W]_{α}, \\ or \\ (ii) & [V]_{α} = [U]_{α} {[W]_{α}}^{- 1} . \end{matrix} \end{matrix}$ If W is a proper fuzzy number (W_l,α is nondecreasing, W_r,α is nonincreasing, W_l,1 ≤ W_r,1).

Remark 2.1. According to [17], if $U \div_{g} V = W \in R_{F}^{+}$ , then two cases are possible

Case I. if U_l,αV_r,α ≤ U_r,αV_l,α, ∀ α ∈ [0, 1] , then $W_{l, α} = \frac{U_{l, α}}{V_{l, α}}, W_{r, α} = \frac{U_{r, α}}{V r, α},$

Case II. if U_l,αV_r,α ≥ U_r,αV_l,α, ∀ α ∈ [0, 1] , then $W_{l, α} = \frac{U_{r, α}}{V_{r, α}}, W_{r, α} = \frac{U_{l, α}}{V_{l, α}} .$

The definition of the boundedness and persistence is as follows (see [11, 12]).

A positive sequence of fuzzy numbers {x_n} is bounded (resp. persistence) provided that there is a Q > 0 (resp. P > 0) such that

$supp x_{n} \subset (0, Q] (resp . supp x_{n} \subset [P, \infty)), n = 1, 2, \dots,$

A positive sequence of fuzzy numbers {x_n} is bounded and persistence provided that there exist P > 0, Q > 0 such that $supp x_{n} \subset [P, Q], n = 1, 2, \dots .$

A positive sequence of fuzzy numbers {x_n} , n = 1, 2, ⋯, is unbounded provided that the norm ∥x_n ∥ , n = 1, 2, ⋯ , is an unbounded sequence.

x_n is called a positive fuzzy solution of FDE (1) provided that {x_n} is a positive sequence of fuzzy numbers satisfying FDE (1).

$x \in R_{F}^{+}$ is a positive equilibrium of FDE (1) provided that $x = \frac{A + x}{B + x} .$

Let $x \in R_{F}^{+}$ and {x_n} be a positive sequence of fuzzy numbers, x_n converges to x with respect to D as n→ ∞ provided that $lim_{n \to \infty} D (x_{n}, x) = 0$ .

Lemma 2.1. [6] If f : R⁺ × R⁺ × R⁺ × R⁺ → R⁺ is continuous, $A, B, C, D \in R_{F}^{+}$ . Then, for α ∈ (0, 1] , $[f (A, B, C, D)]_{α} = f ([A]_{α}, [B]_{α}, [C]_{α}, [D]_{α}) .$ (2)

Lemma 2.2. [5] Let $u \in R_{F}$ with [u] _α = [u_l (α) , u_r (α)] , α ∈ (0, 1] . Then u_l (α) , u_r (α) can be regarded as functions on (0, 1], which satisfy

(i) u_l (α) is nondecreasing and left continuous;

(ii) u_r (α) is nonincreasing and left continuous;

(iii) u_l (1) ≤ u_r (1).

Conversely, for any functions h (α) and g (α) defined on (0, 1] which satisfy (i)-(iii) in the above, there exists a unique $u \in R_{F}$ such that [u] _α = [h (α) , g (α)] for any α ∈ (0, 1].

3 Large time behavior

In the section, to find large time behavior of positive fuzzy solution to FDE (1), we first give the existence of positive fuzzy solution of FDE (1) by virtue of Lemma 2.1.

Theorem 3.1. Consider FDE (1), if the initial values $x_{i} \in R_{F}^{+} (i = - 1, 0)$ and $A, B \in R_{F}^{+}$ , then FDE (1) has a unique positive fuzzy solution x_n.

Proof. Suppose that there exists a sequence of positive fuzzy numbers {x_n} satisfying Eq. (1) with the initial values x_i, i = -1, 0 . Consider the α-level sets, α ∈ (0, 1] , $[x_{n}]_{α} = [x_{n, l, α}, x_{n, r, α}], n = 0, 1, 2, \dots .$ (3) From (1), applying Lemma 2.1, one has that $\begin{matrix} [x_{n + 1}]_{α} & = & [x_{n + 1, l, α}, x_{n + 1, r, α}] \\ = & {[\frac{A + x_{n}}{B + x_{n - 1}}]}_{α} = \frac{[A]_{α} + [x_{n}]_{α}}{[B]_{α} + [x_{n - 1}]_{α}} \\ = & \frac{[A_{l, α} + x_{n, l, α}, A_{r, α} + x_{n, r, α}]}{[B_{l, α} + x_{n - 1, l, α}, B_{r, α} + x_{n - 1, r, α}]} . \end{matrix}$

Noting Remark 2.1, utilizing g-division of fuzzy numbers, One of the following two cases may occur.

Case I $\begin{matrix} [x_{n + 1}]_{α} = [x_{n + 1, l, α}, x_{n + 1, r, α}] \\ = & [\frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}}, \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}] . \end{matrix}$ (4)

Case II $\begin{matrix} [x_{n + 1}]_{α} = [x_{n + 1, l, α}, x_{n + 1, r, α}] \\ = & [\frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}, \frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}}] . \end{matrix}$ (5) If Case I holds true, from (4), one gets, for α ∈ (0, 1] , n ∈ {0, 1, 2, ⋯} , ${\begin{matrix} x_{n + 1, l, α} = \frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}} \\ x_{n + 1, r, α} = \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}} \end{matrix}$ (6)

It is clear that there is a unique solution (x_n,l,α, x_n,r,α) for any initial values (x_k,l,α, x_k,r,α) , k = -1, 0, α ∈ (0, 1].

We only need to show that, for α ∈ (0, 1] , n ∈ N⁺, [x_n,l,α, x_n,r,α] makes certain the positive fuzzy solution x_n of FDE (1) with the initial values x_i (i = 0, - 1), and satisfying (3).

From [8], since $x_{j} \in R_{F}^{+} (j = - 1, 0)$ , for α_i ∈ (0, 1] (i = 1, 2) and α₁ ≤ α₂, it has, for j = -1, 0, $0 < x_{j, l, α_{1}} \leq x_{j, l, α_{2}} \leq x_{j, r, α_{2}} \leq x_{j, r, α_{1}} .$ (7) We claim that, for n = 0, 1, 2, ⋯ , $x_{n, l, α_{1}} \leq x_{n, l, α_{2}} \leq x_{n, r, α_{2}} \leq x_{n, r, α_{1}} .$ (8) By mathematical induction. From (7), for n = -1, 0, it is true. Suppose that, for n ≤ k, k ∈ N⁺, (8) holds true, then it follows from (6) and (8) that, for n = k + 1, $\begin{matrix} x_{k + 1, l, α_{1}} & = & \frac{A_{l, α_{1}} + x_{k, l, α_{1}}}{B_{l, α_{1}} + x_{k - 1, l, α_{1}}} \\ \leq & \frac{A_{l, α_{2}} + x_{k, l, α_{2}}}{B_{l, α_{2}} + x_{k - 1, l, α_{2}}} = x_{k + 1, l, α_{2}} \\ \leq & \frac{A_{r, α_{2}} + x_{k, r, α_{2}}}{B_{r, α_{2}} + x_{k - 1, r, α_{2}}} = x_{k + 1, r, α_{2}} \\ \leq & \frac{A_{r, α_{1}} + x_{k, r, α_{1}}}{B_{r, α_{1}} + x_{k - 1, r, α_{1}}} = x_{k + 1, r, α_{1}} \end{matrix}$ So (8) holds true. Also from (6), one has, for α ∈ (0, 1], ${\begin{matrix} x_{1, l, α} = \frac{A_{l, α} + x_{0, l, α}}{B_{l, α} + x_{- 1, l, α}} \\ x_{1, r, α} = \frac{A_{r, α} + x_{0, r, α}}{B_{r, α} + x_{- 1, r, α}} \end{matrix}$ (9) Since $x_{j} \in R_{F}^{+}, j = - 1, 0$ , and $A, B \in R_{F}^{+}$ , thus, it gets from (9) that x_1,l,α, x_1,r,α are also left continuous. By mathematical induction, we can show that, for n ≥ 1, x_n,l,α and x_n,r,α are left continuous.

Secondly, we will show that $supp x_{n} = \bar{⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}]}$ is compact. It need to prove that ⋃_α∈(0,1] [x_n,l,α, x_n,r,α] is bounded. Since $x_{j} \in R_{F}^{+}, j = - 1, 0$ , and $A, B \in R_{F}^{+}$ , there exist positive real numbers P, Q, K, L, M_j, N_j, j = -1, 0, such that, for α ∈ (0, 1], ${\begin{matrix} [A_{l, α}, A_{r, α}] \subset [P, Q], \\ [B_{l, α}, B_{r, α}] \subset [K, L], \\ [x_{j, l, α}, x_{j, r, α}] \subset [M_{j}, N_{j}] . \end{matrix}$ (10)

Hence from (9) and (10), one has, for n = 1, α ∈ (0, 1], $[x_{1, l, α}, x_{1, r, α}] \subset [\frac{P + M_{0}}{L + N_{- 1}}, \frac{Q + N_{0}}{K + M_{- 1}}] .$ (11) It follows from (11) that, for α ∈ (0, 1], $⋃_{α \in (0, 1]} [x_{1, l, α}, x_{1, r, α}] \subset [\frac{P + M_{0}}{L + N_{- 1}}, \frac{Q + N_{0}}{P + M_{- 1}}] .$ (12) So $\bar{⋃_{α \in (0, 1]} [x_{1, l, α}, x_{1, r, α}]}$ is compact, and $\bar{⋃_{α \in (0, 1]} [x_{1, l, α}, x_{1, r, α}]} \subset (0, \infty)$ . By mathematical induction, one has that $\bar{⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}]}$ is compact, moreover, for n = 1, 2, ⋯ , $\bar{⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}]} \subset (0, \infty) .$ (13) And since x_n,l,α, x_n,r,α are left continuous, from (8) and (13), we get that [x_n,l,α, x_n,r,α] makes certain a positive sequence x_n satisfying (3).

Next we will show that x_n is the solution of FDE (1) with the initial values $x_{i} \in R_{F}^{+}, i = 0, - 1$ , Since, for α ∈ (0, 1], $\begin{matrix} [x_{n + 1}]_{α} & = & [x_{n + 1, l, α}, x_{n + 1, r, α}] \\ = & [\frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}}, \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}] \\ = & {[\frac{A + x_{n}}{B + x_{n - 1}}]}_{α} . \end{matrix}$ Therefore, x_n is the solution of FDE (1) with the initial values x_i, i = 0, - 1.

Suppose that, ${\bar{x}}_{n}$ is another solution of FDE (1) with the initial values $x_{i} \in R_{F}^{+}, i = 0, - 1$ . Then, deducing as above, one has, for n ∈ N⁺, α ∈ (0, 1] . $[{\bar{x}}_{n}]_{α} = [x_{n, l, α}, x_{n, r, α}] .$ (14) It implies from (3) and (14) that $[x_{n}]_{α} = [{\bar{x}}_{n}]_{α}, α \in (0, 1], n \in N^{+},$ so $x_{n} = {\bar{x}}_{n} .$

If Case II occurs, The proof is similar to the proof above. This completes the proof of Theorem 3.1.

In fact, to obtain the results on large time behaviors of positive fuzzy solution for FDE (1). Case I and Case II are considered respectively.

3.1 Case I

In this subsection, using g-division of fuzzy numbers, we are concerned with large time behaviors of positive fuzzy solution to FDE (1) under Case I. To obtain the theoretic results, firstly, we give the following lemma.

Lemma 3.1 Consider the crisp difference equation $y_{n + 1} = \frac{p + y_{n}}{a + y_{n - 1}}, n = 1, 2, \dots,$ (15) where p > 0 and the initial values y_i > 0, i = -1, 0, if $a > 1,$ (16) then these propositions are true.

(i) Every positive solution y_n of (15) satisfies $\frac{a}{a + \frac{p}{a - 1} + y_{0}} \leq y_{n} \leq \frac{p}{a - 1} + y_{0} .$ (17) (ii) The system has a unique positive equilibrium $\bar{y} = \frac{1}{2} (1 - a + \sqrt{(a - 1)^{2} + 4 p})$ which is globally asymptotically stable.

Proof. (i) Let y_n be a positive solution of (15), one gets, for n ≥ 0, $y_{n + 1} = \frac{p + y_{n}}{a + y_{n - 1}} \leq \frac{p}{a} + \frac{1}{a} y_{n} .$ (18) From which, since a > 1, we have $y_{n} \leq \frac{p}{a - 1} + \frac{1}{a^{n}} (y_{0} - \frac{p}{a - 1}) < \frac{p}{a - 1} + y_{0} .$ (19) It follows from (15) and (19) that $\frac{a}{a + \frac{p}{a - 1} + y_{0}} \leq y_{n}$ (20) The proof of (i) is completed.

(ii) From (15), it is easy to obtain that the unique positive equilibrium of (15) is $\bar{y} = \frac{1}{2} (1 - a + \sqrt{(a - 1)^{2} + 4 p}) .$ The linearized equation of system (15) at $\bar{y}$ is $y_{n + 1} - {ry}_{n} + {sy}_{n - 1} = 0,$ (21) where $r = \frac{2}{a + 1 + \sqrt{(a - 1)^{2} + 4 p}},$ $s = \frac{1 - a + \sqrt{(a - 1)^{2} + 4 p}}{a + 1 + \sqrt{(a - 1)^{2} + 4 p}} .$ Since a > 1, it is easy to see that |r|<1 - s < 2, then two roots of characteristic equation lie in the unit circle. Thus the equilibrium point $\bar{y}$ is locally asymptotically stable.

From (17), suppose that $lim_{n \to \infty} sup y_{n} = Λ, lim_{n \to \infty} inf y_{n} = λ$ (22) where Λ, λ ∈ (0, ∞) . Then from (15) we have $Λ \leq \frac{p + Λ}{a + λ}, λ \geq \frac{p + λ}{a + Λ} .$ From which, one has a (Λ - λ) ≤ Λ - λ. Since a > 1, so Λ = λ . Namely $lim_{n \to \infty} y_{n} = \bar{y} .$ Therefore the equilibrium point $\bar{y}$ is globally asymptotically stable.

Theorem 3.2 Consider FDE (1), where the parameters $A, B \in R_{F}^{+}$ , and the initial values $x_{i} \in R_{F}^{+}, i = - 1, 0$ . If there exists a real constant K satisfying $B_{l, α} \geq K > 1, α \in (0, 1],$ (23) and for α ∈ (0, 1] , n = 0, 1, 2, ⋯ , $\begin{matrix} (A_{l, α} + x_{n, l, α}) (B_{r, α} + x_{n - 1, r, α}) \\ \leq & (A_{r, α} + x_{n, r, α}) (B_{l, α} + x_{n - 1, l, α}), \end{matrix}$ (24) then the following propositions are true.

(i) Every positive fuzzy solution x_n of FDE (1) is bounded and persistent.

(ii) FDE (1) has a unique positive equilibrium $\bar{x}$ .

(iii) Every positive fuzzy solution x_n of FDE (1) tends to the positive equilibrium $\bar{x}$ as n → ∞ . Moreover the unique positive equilibrium $\bar{x}$ is globally asymptotically stable.

Proof. (i) Let x_n be a positive fuzzy solution of FDE (1). Since (10) and (24) hold true, according to Lemma 3.1, from (6), one has $\begin{matrix} \frac{A_{l, α}}{B_{l, α} + \frac{A_{l, α}}{B_{l, α} - 1} + x_{0, l, α}} \leq x_{n, l, α} \\ \leq & \frac{A_{l, α}}{B_{l, α} - 1} + x_{0, l, α}, \end{matrix}$ (25)

$\begin{matrix} \frac{A_{r, α}}{B_{r, α} + \frac{A_{r, α}}{B_{r, α} - 1} + x_{0, r, α}} \leq x_{n, r, α} \\ \leq & \frac{A_{r, α}}{B_{r, α} - 1} + x_{0, r, α} . \end{matrix}$ (26)

From (10), (25),(26), and (3), we have, for n = 1, 2, ⋯ , $[x_{n}]_{α} = [x_{n, l, α}, x_{n, r, α}] \subset [M, N],$ (27) where $M = \frac{P}{L + \frac{Q}{K - 1} + N_{0}}, N = \frac{Q}{K - 1} + N_{0} .$

From (27), it follows that, for n ≥ 1,

$⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}] \subset [M, N] .$ So $\bar{⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}]} \subset [M, N] .$ This completes the proof of Part (i).

(ii) Suppose that (x_l,α, x_r,α) satisfies the following system ${\begin{matrix} x_{l, α} & = & \frac{A_{l, α} + x_{l, α}}{B_{l, α} + x_{l, α}}, \\ x_{r, α} & = & \frac{A_{r, α} + x_{r, α}}{B_{r, α} + x_{r, α}} \end{matrix}$ (28) Then (x_l,α, x_r,α) is given by ${\begin{matrix} x_{l, α} = \frac{1}{2} (1 - B_{l, α} + \sqrt{(B_{l, α} - 1)^{2} + 4 A_{l, α}}), \\ x_{r, α} = \frac{1}{2} (1 - B_{r, α} + \sqrt{(B_{r, α} - 1)^{2} + 4 A_{r, α}}) . \end{matrix}$ (29) Let x_n be a positive fuzzy solution of FDE (1) satisfying (10). Then applying Lemma 3.1, noting (6), one has $lim_{n \to \infty} x_{n, l, α} = x_{l, α}, lim_{n \to \infty} x_{n, r, α} = x_{r, α} .$ (30) From (27) and (30), we have, for 0 < α₁ ≤ α₂ ≤ 1, $0 < x_{l, α_{1}} \leq x_{l, α_{2}} \leq x_{r, α_{2}} \leq x_{r, α_{1}} .$ (31) Since A_l,α, A_r,α, B_l,α, B_r,α are left continuous. From (29) it follows that x_l,α, x_r,α are left continuous. It follows from (10) and (29) that ${\begin{matrix} x_{l, α} \geq c = \frac{1}{2} (1 - L + \sqrt{(K - 1)^{2} + 4 P}), \\ 0 < x_{r, α} \leq d = \frac{1}{2} (1 - K + \sqrt{(L - 1)^{2} + 4 Q}) . \end{matrix}$ (32) Therefore [x_l,α, x_r,α] ⊂ [c, d], and so $⋃_{α \in (0, 1]} [x_{l, α}, x_{r, α}] \subset [c, d] .$ (33) From which it is obvious that ⋃_α∈(0,1] [x_l,α, x_r,α] is compact and $⋃_{α \in (0, 1]} [x_{l, α}, x_{r, α}] \subset (0, \infty) .$ (34) So from Lemma 2.2, (28), (31) and (34), since x_l,α, x_r,α, α ∈ (0, 1] ascertains a fuzzy number $\bar{x}$ such that $\bar{x} = \frac{A + \bar{x}}{B + \bar{x}}, [\bar{x}]_{α} = [x_{l, α}, x_{r, α}], α \in (0, 1] .$ (35) So $\bar{x}$ is a positive equilibrium of FDE (1).

(iii) From (30), one has that $\begin{matrix} lim_{n \to \infty} D (x_{n}, \bar{x}) = lim_{n \to \infty} sup_{α \in (0, 1]} {max {| x_{n, l, α} - x_{l, α} |, \\ | x_{n, r, α} - x_{r, α} |}} . \end{matrix}$ (36) Namely, every positive fuzzy solution x_n of FDE (1) converges the unique equilibrium $\bar{x}$ with respect to D as n → ∞ .

Let ɛ > 0, suppose that $0 < δ < min {ɛ, K + c - 1 - d},$ (37) where c, d are defined as (32).

On the other hand, let x_n be a positive fuzzy solution of FDE (1) satisfying $D (x_{i}, \bar{x}) \leq δ < ɛ, i = - 1, 0 .$ From which, for α ∈ (0, 1] , $| x_{i, l, α} - x_{l, α} | \leq δ, | x_{i, r, α} - x_{r, α} | \leq δ .$ (38) It follows from (6), (10) and (38) that $\begin{matrix} x_{1, l, α} - x_{l, α} = \frac{A_{l, α} + x_{0, l, α}}{B_{l, α} + x_{- 1, l, α}} - x_{l, α} \\ \leq & \frac{A_{l, α} + x_{l, α} + δ}{B_{l, α} + x_{l, α} - δ} - x_{l, α} \\ = & δ \frac{1 + x_{l, α}}{B_{l, α} + x_{l, α} - δ} \leq δ \frac{1 + x_{r, α}}{K + x_{l, α} - δ}, \end{matrix}$ (39) $\begin{matrix} x_{1, r, α} - x_{r, α} = \frac{A_{r, α} + x_{0, r, α}}{B_{r, α} + x_{- 1, r, α}} - x_{r, α} \\ \leq & \frac{A_{r, α} + x_{r, α} + δ}{B_{r, α} + x_{r, α} - δ} - x_{r, α} \\ = & δ \frac{1 + x_{r, α}}{B_{r, α} + x_{r, α} - δ} \leq δ \frac{1 + d}{K + c - δ} \end{matrix}$ (40) From (37),(39) and (40), one has that $| x_{1, l, α} - x_{l, α} | < δ < ɛ, | x_{1, r, α} - x_{r, α} | < δ < ɛ .$ (41) Working inductively,it is easy to get, for n ≥ 1, α ∈ (0, 1], $| x_{n, l, α} - x_{l, α} | < ɛ, | x_{n, r, α} - x_{r, α} | < ɛ .$ (42) So $D (x_{n}, \bar{x}) < ɛ$ . Namely, the positive equilibrium $\bar{x}$ is locally stable. From (36), thus the equilibrium $\bar{x}$ is globally asymptotically stable. The proof of Theorem 3.2 is completed.

3.2 Case II

Secondly, if Case II holds true, from (5), then (x_n,l,α, x_n,r,α) satisfies the following family of systems of parametric crisp difference equations, for α ∈ (0, 1] , n = 0, 1, 2, ⋯ , ${\begin{matrix} x_{n + 1, l, α} = \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}, \\ x_{n + 1, r, α} = \frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}} . \end{matrix}$ (43)

In order to find large time behaviors of FDE (1), we give the following lemma.

Lemma 3.2. Consider the system of crisp difference equations, for n = 0, 1, ⋯ , $y_{n + 1} = \frac{q + z_{n}}{b + z_{n - 1}}, z_{n + 1} = \frac{p + y_{n}}{a + y_{n - 1}},$ (44) where a, b, p, q, y_i, z_i ∈ (0, + ∞) , i = -1, 0. If $b > a > 1,$ (45) then the following propositions are true.

(i) Every positive solution (y_n, z_n) of (44) is persistence and bounded.

(ii) The system exists a unique positive equilibrium $\begin{matrix} \bar{y} & = & \frac{q - p + 1 - ab}{2 (b + 1)} \\ + & \frac{\sqrt{(ab + p - q - 1)^{2} + 4 (b + 1) (aq + p)}}{2 (b + 1)}, \end{matrix}$ (46) $\begin{matrix} \bar{z} & = & \frac{p - q + 1 - ab}{2 (a + 1)} \\ + & \frac{\sqrt{(ab + q - p - 1)^{2} + 4 (a + 1) (bp + q)}}{2 (a + 1)} . \end{matrix}$ (47) Moreover, it is globally asymptotically stable.

Proof. (i) From (44), one has $\begin{matrix} y_{n} & = & \frac{q + z_{n - 1}}{b + z_{n - 2}} \leq \frac{q}{b} + \frac{1}{b} z_{n - 1} \\ = \frac{q}{b} + \frac{1}{b} (\frac{p + y_{n - 2}}{a + y_{n - 3}}) \\ \leq & \frac{q}{b} + \frac{p}{ab} + \frac{1}{ab} y_{n - 2} \\ \leq & \frac{q}{b} + \frac{p}{ab} + \frac{1}{ab} (\frac{q}{b} + \frac{1}{b} z_{n - 3}) \\ \leq & \frac{q}{b} + \frac{p}{ab} + \frac{q}{{ab}^{2}} + \frac{p}{a^{2} b^{2}} + \frac{1}{a^{2} b^{2}} y_{n - 4} \leq \dots \\ \leq & (\frac{q}{b} + \frac{q}{{ab}^{2}} + \dots + \frac{q}{a^{k} b^{k}}) \\ + (\frac{p}{ab} + \frac{p}{a^{2} b^{2}} + \dots + \frac{p}{a^{k} b^{k}}) + \frac{1}{a^{k} b^{k}} y_{n - 2 k} \\ = & \frac{\frac{q}{b} [1 - {(\frac{1}{ab})}^{k}]}{1 - \frac{1}{ab}} + \frac{\frac{p}{ab} [1 - {(\frac{1}{ab})}^{k}]}{1 - \frac{1}{ab}} \\ + \frac{1}{(ab)^{k}} y_{n - 2 k} \\ \leq & \frac{aq}{ab - 1} + \frac{p}{ab - 1} + y_{n - 2 k} \end{matrix}$ (48) It follows from (48) that $y_{n} \leq Θ : = {\begin{matrix} \frac{aq}{ab - 1} + \frac{p}{ab - 1} + y_{0}, & n = 2 k, \\ \frac{aq}{ab - 1} + \frac{p}{ab - 1} + y_{1}, & n = 2 k + 1 . \end{matrix}$ (49) Similarly, we have $z_{n} \leq H : = {\begin{matrix} \frac{bp}{ab - 1} + \frac{q}{ab - 1} + z_{0}, & n = 2 k, \\ \frac{bp}{ab - 1} + \frac{q}{ab - 1} + z_{1}, & n = 2 k + 1 . \end{matrix}$ (50) From (44), (49), and (50), it is easy to see that $y_{n} \geq Γ, z_{n} \geq ϒ, n \geq 1 .$ (51) where $Γ = \frac{q}{b + H}, ϒ = \frac{p}{a + Θ} .$ And so the positive solution (y_n, z_n) of (44) is persistence and bounded.

(ii) From (44), one has that the unique positive equilibrium $(\bar{y}, \bar{z})$ is given by (46) and (47). The linearized equation of system (44) at $(\bar{y}, \bar{z})$ is $Ψ_{n + 1} = G Ψ_{n} .$ (52) In which $\begin{matrix} Ψ_{n} = (\begin{matrix} y_{n} \\ y_{n - 1} \\ z_{n} \\ z_{n - 1} \end{matrix}), \\ G = (\begin{matrix} 0 & 0 & \frac{1}{b + \bar{z}} & - \frac{\bar{y}}{b + \bar{z}} \\ 1 & 0 & 0 & 0 \\ \frac{1}{a + \bar{y}} & - \frac{\bar{z}}{a + \bar{y}} & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}) . \end{matrix}$ Let λ_i, (i = 1, 2, 3, 4) , be the eigenvalues of G, and Ω = diag (m₁, m₂, m₃, m₄) be a diagonal matrix, where m₁ = m₃ = 1, m_k = 1 - kɛ (k = 2, 4), and

$\begin{matrix} 0 < ɛ & < & min {\frac{1}{4} (1 - \frac{\bar{y}}{b + \bar{z} - 1}), \\ \frac{1}{2} (1 - \frac{\bar{z}}{a + \bar{y} - 1})} . \end{matrix}$ (53) Clearly, Ω is invertible. Computing ΩGΩ^-1, one has $Ω G Ω^{- 1} = (\begin{matrix} 0 & 0 & \frac{m_{1} m_{3}^{- 1}}{b + \bar{z}} & - \frac{m_{1} m_{4}^{- 1} \bar{y}}{b + \bar{z}} \\ m_{2} m_{1}^{- 1} & 0 & 0 & 0 \\ \frac{m_{3} m_{1}^{- 1}}{a + \bar{y}} & - \frac{m_{3} m_{2}^{- 1} \bar{z}}{a + \bar{y}} & 0 & 0 \\ 0 & 0 & m_{4} m_{3}^{- 1} & 0 \end{matrix}) .$ (54) Noting (53), One gets from (54) that $\frac{1}{b + \bar{z}} + \frac{\bar{y} m_{4}^{- 1}}{b + \bar{z}} = \frac{1 + \frac{1}{1 - 4 ɛ} \bar{y}}{b + \bar{z}} < 1,$ $\frac{1}{a + \bar{y}} + \frac{\bar{z} m_{2}^{- 1}}{a + \bar{y}} = \frac{1 + \frac{1}{1 - 2 ɛ} \bar{z}}{a + \bar{y}} < 1 .$ Since G has the same eigenvalues as ΩGΩ^-1, one has $\begin{matrix} max_{1 \leq i \leq 4} | λ_{i} | \leq ∥ Ω G Ω^{- 1} ∥_{\infty} = max \\ {m_{2} m_{1}^{- 1}, m_{4} m_{3}^{- 1}, \frac{1 + \bar{y} m_{4}^{- 1}}{b + \bar{z}}, \frac{1 + \bar{z} m_{2}^{- 1}}{a + \bar{y}}} < 1 . \end{matrix}$ This deduces that the positive equilibrium $(\bar{y}, \bar{z})$ of (44) is locally stable.

On the other hand, since y_n, z_n are bounded and persistence, suppose that $lim_{n \to \infty} sup y_{n} = Λ_{1}, lim_{n \to \infty} inf y_{n} = λ_{1},$ $lim_{n \to \infty} sup z_{n} = Λ_{2}, lim_{n \to \infty} inf z_{n} = λ_{2} .$ In which Λ_i, λ_i ∈ (0, ∞) , i = 1, 2 .

Then from (44), one has ${\begin{matrix} Λ_{1} \leq \frac{q + Λ_{2}}{b + λ_{2}}, Λ_{2} \leq \frac{p + Λ_{1}}{a + λ_{1}}, \\ λ_{1} \geq \frac{q + λ_{2}}{b + Λ_{2}}, λ_{2} \geq \frac{p + λ_{1}}{a + Λ_{1}} . \end{matrix}$ (55) It follows from (55) that $(a - 1) (Λ_{1} - λ_{1}) + (b - 1) (Λ_{2} - λ_{2}) \leq 0 .$ (56) Since condition (45) holds true, thus Λ_i = λ_i (i = 1, 2). Namely, one has $lim_{n \to \infty} y_{n} = \bar{y}, lim_{n \to \infty} z_{n} = \bar{z} .$ Therefore the positive equilibrium $(\bar{y}, \bar{z})$ of (44) is globally asymptotically stable. This completes the proof of Lemma 3.3.

Theorem 3.3. Consider FDE (1), where the initial values $x_{i} \in R_{F}^{+}, i = - 1, 0$ , and $A, B \in R_{F}^{+}$ . If, for ∀α ∈ (0, 1],

\begin{matrix} (A_{r, α} + x_{n, r, α}) (B_{l, α} + x_{n - 1, l, α}) \\ \leq & (A_{l, α} + x_{n, l, α}) (B_{r, α} + x_{n - 1, r, α}) \end{matrix}

(57)

Furthermore, (10) and (23) hold, then the following propositions are true.

(i) Every positive fuzzy solution x_n of FDE (1) is bounded and persistent.

(ii) FDE (1) has a unique positive equilibrium $\bar{x}$ .

(iii) Every positive fuzzy solution x_n of FDE (1) tends to the positive equilibrium $\bar{x}$ with respect to D as n → ∞ .

Proof. (i) Let x_n be a positive fuzzy solution of FDE (1). By virtue of Lemma 3.2, one has

$\begin{matrix} \frac{A_{r, α}}{B_{l, α} + \frac{B_{r, α} A_{l, α} + A_{r, α}}{B_{l, α} B_{r, α} - 1} + max {x_{0, l, α}, x_{- 1, l, α}}} \\ \leq & x_{n, l, α} \leq \frac{B_{l, α} A_{r, α} + A_{r, α}}{B_{l, α} B_{r, α} - 1} \\ + max {x_{0, l, α}, x_{- 1, l, α}}, \end{matrix}$ (58)

$\begin{matrix} \frac{A_{l, α}}{B_{r, α} + \frac{B_{l, α} A_{r, α} + A_{l, α}}{B_{l, α} B_{r, α} - 1} + max {x_{0, r, α}, x_{- 1, r, α}}} \\ \leq & x_{n, r, α} \leq \frac{B_{r, α} A_{l, α} + A_{l, α}}{B_{l, α} B_{r, α} - 1} \\ + max {x_{0, r, α}, x_{- 1, r, α}} . \end{matrix}$ (59) From (5) and (26), we have $[x_{n}]_{α} = [x_{n, l, α}, x_{n, r, α}] \subset [W, T]$ (60) where $W = \frac{P}{L + \frac{LQ + Q}{K^{2} - 1} + max {N_{0}, N_{- 1}}},$ $T = \frac{LQ + Q}{K^{2} - 1} + max {N_{0}, N_{- 1}} .$ Therefore $\bar{⋃_{α \in (0, 1]} [x_{n, l, α}, x_{n, r, α}]} \subseteq [W, T] .$ Namely the positive fuzzy solution is bounded and persistent.

(ii) Suppose that the positive fuzzy number $\bar{x}$ satisfies (43). One has for α ∈ (0, 1] , $x_{l, α} = \frac{A_{r, α} + x_{r, α}}{B_{r, α} + x_{r, α}}, x_{r, α} = \frac{A_{l, α} + x_{l, α}}{B_{l, α} + x_{l, α}} .$ (61) From (61), we have $\begin{matrix} x_{l, α} = \frac{- u + \sqrt{u^{2} + 4 v}}{2 (B_{r, α} + 1)}, x_{r, α} = \frac{- β + \sqrt{β^{2} + 4 η}}{2 (B_{l, α} + 1)} . \end{matrix}$ where $u = B_{l, α} B_{r, α} + A_{l, α} - A_{r, α} - 1,$ $v = (B_{r, α} + 1) (B_{l, α} A_{r, α} + A_{l, α}),$ $β = (B_{l, α} B_{r, α} + A_{r, α} - A_{l, α} - 1,$ $η = (B_{l, α} + 1) (B_{r, α} A_{l, α} + A_{r, α}) .$

It is easy to show that x_l,α, x_r,α, α ∈ (0, 1] makes certain a positive fuzzy number $\bar{x}$ satisfying (35). So $\bar{x}$ is a positive equilibrium of FDE (1).

(iii) Let x_n be a positive fuzzy solution of FDE (1) such that Case II occurs. i.e. (43) are true. Since B_l,α > K > 1, using Lemma 3.2 to system (43), one has $lim_{n \to \infty} x_{n, l, α} = x_{l, α}, lim_{n \to \infty} x_{n, r, α} = x_{r, α},$ (62) Therefore from (62), one gets $\begin{matrix} lim_{n \to \infty} D (x_{n}, x) & = & lim_{n \to \infty} sup_{α \in (0, 1]} {max {| x_{n, l, α} - x_{l, α} |, \\ | x_{n, r, α} - x_{r, α} |}} = 0 . \end{matrix}$ The proof of Theorem 3.3 is completed.

Remark 3.1. Diseases diagnosis often involves several degree of uncertainty or imprecision, especially in epidemiological problem. It is natural to study short or large time behaviors of epidemic model in fuzzy dynamical system ([see [14, 23, 25]). In the past decades, study on asymptotic behavior of fuzzy dynamical system is a very new research field. Its fundamental idea is to take classical modelled by difference equation, differential equation to extend fuzzy set theoretic framework. Therefore FDE or fuzzy differential equation is a proper tool to study fuzzy dynamical system. To find dynamical behaviors of epidemic model with fuzzy parameter, fuzzy variable, or fuzzy initial conditions. It is very important topic worth studying corresponding fuzzy difference (or differential) equation.

4 Numerical examples

In this section, we give the following two examples to verify the correctness of theoretic results.

Example 4.1 Consider FDE (1), we take $A, B \in R_{F}^{+}$ and the initial values $x_{i} \in R_{F}^{+}, i = - 1, 0$ , whose membership functions are described as follows $A (t) = {\begin{matrix} t - 0.5, & 0.5 \leq t \leq 1.5 \\ - t + 2.5, & 1.5 \leq t \leq 2.5 \end{matrix}$ (63) $B (t) = {\begin{matrix} 2 t - 4, & 2 \leq t \leq 2.5 \\ - 2 t + 6, & 2.5 \leq t \leq 3 . \end{matrix}$ (64) $x_{- 1} (t) = {\begin{matrix} t - 1, & 1 \leq t \leq 2 \\ - t + 3, & 2 \leq t \leq 3 \end{matrix}$ (65) $x_{0} (t) = {\begin{matrix} t - 3, & 3 \leq t \leq 4 \\ - t + 5, & 4 \leq t \leq 5 \end{matrix}$ (66) From (63) and (64), one has $[A]_{α} = [0.5 + α, 2.5 - α], α \in (0, 1] .$ (67) $[B]_{α} = [2 + \frac{1}{2} α, 3 - \frac{1}{2} α], α \in (0, 1] .$ (68) From (65) and (66), one has $[x_{- 1}]_{α} = [1 + α, 3 - α], α \in (0, 1] .$ (69) $[x_{0}]_{α} = [3 + α, 5 - α], α \in (0, 1] .$ (70) Therefore, it follows that $\bar{⋃_{α \in (0, 1]} [A]_{α}} = [0.5, 2.5], \bar{⋃_{α \in (0, 1]} [B]_{α}} = [2, 3],_$ (71) $\bar{⋃_{α \in (0, 1]} [x_{- 1}]_{α}} = [1, 3], \bar{⋃_{α \in (0, 1]} [x_{0}]_{α}} = [3, 5] .$ (72) From FDE (1), it can turn into the following crisp difference equation system with parameter α ∈ (0, 1] , ${\begin{matrix} x_{n + 1, l, α} = \frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}}, \\ x_{n + 1, r, α} = \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}, \end{matrix}$ (73) Therefore, B_l,α > 1, ∀ α ∈ (0, 1] , and $x_{i} \in R_{F}^{+}, i = - 1, 0,$ By Theorem 3.2, one has that every positive solution x_n of FDE (1) is persistence and bounded. and there exists a unique positive fuzzy equilibrium $\bar{x} = (0.3660, 0.6861, 0.8708)$ . Meanwhile, every positive fuzzy solution x_n of FDE (1) tends to $\bar{x}$ as n → ∞ . (see Figs. 1–3)

Fig. 1

The large time behavior of FDE (1).

Fig. 2

The positive solution of system (73) at α = 0 and α = 0.25 .

Fig. 3

The positive solution of system (73) at α = 0.75 and α = 1 .

Example 4.2 Consider FDE (1). where x₀ is taken as (66), $A, B, x_{- 1} \in R_{F}^{+}$ , whose membership functions are as follows

$A (t) = {\begin{matrix} 2 t - 3, & 1.5 \leq t \leq 2 \\ - 2 t + 5, & 2 \leq t \leq 2.5 \end{matrix},$ (74) $x_{- 1} (t) = {\begin{matrix} 0.5 t - 0.5, & 1 \leq t \leq 3 \\ - 0.5 t + 2.5, & 3 \leq t \leq 5 \end{matrix}$ (75)

$B (t) = {\begin{matrix} 0.5 t - 1, & 2 \leq t \leq 4 \\ - 0.5 t + 3, & 4 \leq t \leq 6 \end{matrix}$ (76)

From (74), (75) and (76), one has $[A]_{α} = [1.5 + \frac{1}{2} α, 2.5 - \frac{1}{2} α], α \in (0, 1] .$ (77) $[x_{- 1}]_{α} = [1 + 2 α, 5 - 2 α], α \in (0, 1] .$ (78) $[B]_{α} = [2 + 2 α, 6 - 2 α], α \in (0, 1] .$ (79) Therefore, it follows that $\bar{⋃_{α \in (0, 1]} [A]_{α}} = [1.5, 2.5], \bar{⋃_{α \in (0, 1]} [B]_{α}} = [2, 6],$ (80) $\bar{⋃_{α \in (0, 1]} [x_{- 1}]_{α}} = [1, 5],$ (81) From FDE (1), it can turn into the following crisp difference equation system with parameter α ∈ (0, 1] , ${\begin{matrix} x_{n + 1, l, α} = \frac{A_{r, α} + x_{n, r, α}}{B_{r, α} + x_{n - 1, r, α}}, \\ x_{n + 1, r, α} = \frac{A_{l, α} + x_{n, l, α}}{B_{l, α} + x_{n - 1, l, α}}, \end{matrix}$ (82) It is clear that (23) and (57) are satisfied and $x_{i} \in R_{F}^{+}$ , by Theorem 3.3, FDE (1) has a unique positive fuzzy equilibrium $\bar{x} = (0.4852, 0.5616, 0.7988)$ . Moreover every positive fuzzy solution x_n of FDE (1) tends to $\bar{x}$ D as n → ∞ . (see Figs. 4–6)

Fig. 4

The large time behavior of FDE (1).

Fig. 5

The positive solution of system (82) at α = 0 and α = 0.25.

Fig. 6

The positive solution of system (82) at α = 0.75 and α = 1.

5 Conclusion and future work

In this paper, using g-division of fuzzy number, we explored large time behaviors of second order fractal FDE $x_{n + 1} = \frac{A + x_{n}}{B + x_{n - 1}}$ . The results obtained are as follows

(1) Provided that B_l,α > 1, and (A_l,α + x_n,l,α) (B_r,α + x_n-1,r,α) ≤ (A_r,α + x_n,r,α) (B_l,α + x_n-1,l,α) , α ∈ (0, 1]. Case I occurs, then the positive fuzzy solution x_n is bounded and persistence, and it tends to the unique equilibrium x as n → ∞ .

(2) Provided that B_l,α > 1, and (A_r,α + x_n,r,α) (B_l,α + x_n-1,l,α) ≤ (A_l,α + x_n,l,α) (B_r,α + x_n-1,r,α) , α ∈ (0, 1]. Case II occurs, then the positive fuzzy solution x_n is bounded and persistence, and it tends to the unique equilibrium x as n → ∞ .

Up to now, due to lack of available theoretical tools, some complex dynamic behaviors of fuzzy discrete model have not been reported such as oscillation property, the existence of periodic solution and bifurcation. These dynamical behaviors are worth further discussion. We will explore the research on large time behavior of discrete population model or discrete economical model with fuzzy uncertainty conditions.

Footnotes

Acknowledgment

This work was financially supported by Guizhou Scientific and Technological Platform Talents (GCC [2022]020-1), Scientific Research Foundation of Guizhou Provincial Department of Science and Technology ([2022]021, [2022]026), and Scientific Climbing Programme of Xiamen University of Technology (XPDKQ20021), Postgraduate Research Foundation of Guizhou University of Finance and Economics(2022ZXSY139),Scientific Research Foundation of Guizhou University of Finance and Economics(2022ZCZX076).

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