1 Introduction
Difference equation is one of the most important dynamical model. It is the analogue of corresponding differential equation and delay differential equation having an extensive applications in computer science, control engineering, chemistry, biology, economics, etc. (see [1‐11]). Recently, many authors are interested in studying qualitative features of rational difference equation. For example, Bešo et al. [12] proposed a second-order rational difference equation with quadratic termwhere \(\gamma>0, \delta >0\), and the initial values \(x_0>0, x_{-1}>0\).
$$\begin{aligned} x_{n+1}=\gamma +\delta \frac{x_n}{x_{n-1}^2}, n=0,1,\ldots , \end{aligned}$$
(1)
In 2017, Khyat et al. [13] explored a similar model with quadratic term in both the numerator and denominatorFurthermore, they obtained the globally asymptotically stability of (2) and the direction of the Neimark–Sacker bifurcation.
$$\begin{aligned} x_{n+1}=a+ \frac{x_n^2}{x_{n-1}^2}, n=0,1,\ldots . \end{aligned}$$
(2)
Anzeige
In the last few decades, there are many publications on the stability, oscillatory, periodicity, and boundedness of nonlinear rational difference equations. Moreover a lot of similar qualitative features also appear in nonlinear rational difference equation systems (see [14‐17]).
Although these models are very simple in their forms, we can not understand fully the qualitative features of their solutions. In fact, these models inevitably implicit inherent uncertainty or vague. It is well known that fuzzy set is a powerful tool to cope with these uncertainties or subjective information in mathematical model. Therefore, it is a natural method to explore dynamical model with uncertainty or impression by establishing fuzzy difference equation (FDE) or fuzzy differential equation.
FDE is a special kind of difference equation whose coefficients and the initial condition are fuzzy numbers, and its’ solution is a sequence of fuzzy number. Due to the advantage of FDE in dealing with inherent imprecision, the study on qualitative features of these models has become an important research topic both from theoretical viewpoint and in applications. Therefore, in the last decades, there has been an increasing results on the study of FDE (see [18‐33]). It is found that fuzzy set theory has potential in the application of fuzzy differential equations and fuzzy time series (see [34‐40]).
Inspired by previous works, in this paper, utilizing a generalization of division (g-division) of fuzzy numbers, we explore the qualitative features of positive fuzzy solution to the following FDE with quadratic termwhere the initial condition \(x_{-1}, x_0\), and the parameters A, B are positive fuzzy numbers.
$$\begin{aligned} x_{n+1}=A+\frac{Bx_n}{x_{n-1}^2},\ n=0,1,\ldots , \end{aligned}$$
(3)
Anzeige
The organization of this paper is arranged as follows. Section 2 gives some basic concepts of fuzzy numbers used throughout the paper. In Sect. 3, the qualitative features of the positive fuzzy solution to FDE (3) are obtained by virtue of g-division of fuzzy numbers. Section 4 presents two illustrative examples to verify our theoretic results. A general conclusion and discussion are drawn in Sect. 5.
2 Preliminary and definitions
In this section, we first review some basic concepts and definitions to be used in the sequel. These particular descriptions are found in many publications [20‐22].
A function defined as \(U: R\rightarrow [0,1]\) is called a fuzzy number if it is normal, fuzzy convex, upper semi-continuous, and compactly support on R.
For \(\alpha \in (0,1]\), we denote the \(\alpha -\)cuts of U by \([U]_\alpha =\{x\in R: U(x)\ge \alpha \}\), and for \(\alpha =0\), the support of U is written as \(\text{ supp }U=[U]_0=\overline{\{x\in R: U(x)>0\}}\).
It is easy to see that the \([U]_\alpha \) is a closed interval. Provided that \(\text{ supp } U\subset (0,\infty ),\) then U is said to be a positive fuzzy number. Particularly, if U is a positive real number, then it is said to be a trivial fuzzy number, i.e., \([U]_\alpha =[U, U], \alpha \in (0,1].\)
Let U, V be fuzzy numbers, \([U]_\alpha =[U_{l,\alpha },U_{r,\alpha }],[V]_\alpha =[V_{l,\alpha },V_{r,\alpha }],\alpha \in [0,1]\), \(k>0\). Addition and multiplication of fuzzy numbers are defined as follows.The family of fuzzy numbers with addition and multiplication defined by Eqs. (4) and (5) is written as \(\Re _F\). Particularly, the family of positive (resp. negative) fuzzy numbers is denoted by \(\Re _F^+\)( resp. \(\Re _F^-\)).
$$\begin{aligned}{}[U+V]_\alpha= & {} [U_{l,\alpha }+V_{l,\alpha },U_{r,\alpha }+V_{r,\alpha }], \end{aligned}$$
(4)
$$\begin{aligned}{}[kU]_\alpha= & {} [kU_{l,\alpha },kU_{r,\alpha }]. \end{aligned}$$
(5)
Definition 2.1
Let \(U,V\in \Re _F\), the metric is defined as follows.Obviously, \((\Re _F,D)\) is a complete metric space.
$$\begin{aligned} D(U,V)=\sup _{\alpha \in [0,1]}\max \{\mid U_{l,\alpha }-V_{l,\alpha }\mid ,\mid U_{r,\alpha }-V_{r,\alpha }\mid \}. \end{aligned}$$
(6)
Definition 2.2
[41] Let \(U, V\in \Re _F\), \([U]_\alpha =[U_{l,\alpha }, U_{r,\alpha }], [V]_\alpha =[V_{l,\alpha }, V_{r,\alpha }]\), \(0\notin [V]_\alpha , \forall \alpha \in [0,1]\). The g-division \(\div _g\) of the fuzzy numbers is written as \(W=U\div _gV\) with \(\alpha -\)cuts \([W]_\alpha =[W_{l,\alpha },W_{r,\alpha }]\)( \([W]_\alpha ^{-1}=[1/W_{r,\alpha },1/W_{l,\alpha }]\)), whereif W is a proper fuzzy number.
$$\begin{aligned}{}[W]_\alpha =[U]_\alpha \div _g[V]_\alpha \Longleftrightarrow \left\{ \begin{array}{ll} (i) &{} [U]_\alpha =[V]_\alpha [W]_\alpha ,\\ &{}\\ (ii)&{} [V]_\alpha =[U]_\alpha [W]_\alpha ^{-1}, \end{array} \right. \end{aligned}$$
(7)
Remark 2.1
According to [41], Let \(U,V\in \Re _F^+\), if \(U\div _g V=W\in \Re _F^+\) exists, then there are two cases
Case (i). if \(U_{l,\alpha }V_{r,\alpha }\le U_{r,\alpha }V_{l,\alpha },\forall \alpha \in [0,1],\) then \(W_{l,\alpha }=\frac{U_{l,\alpha }}{V_{l,\alpha }}, W_{r,\alpha }=\frac{U_{r,\alpha }}{V{r,\alpha }},\)
Case (ii). if \(U_{l,\alpha }V_{r,\alpha }\ge U_{r,\alpha }V_{l,\alpha },\forall \alpha \in [0,1],\) then \(W_{l,\alpha }=\frac{U_{r,\alpha }}{V_{r,\alpha }}, W_{r,\alpha }=\frac{U_{l,\alpha }}{V{l,\alpha }}.\)
Definition 2.3
Let \(x_n\in \Re _F^+, n=1, 2, \ldots \), \((x_n)\) is said to be persistence (resp. bounded) provided that there is \(M>0\) (resp. \(N>0\)) satisfyingThe sequence \((x_n)\) is bounded and persistence if there are \(M, N>0\) satisfyingThe sequence \((x_n)\) is unbounded if the norm of \(x_n\), \(\Vert x_n\Vert ,n=1,2,\ldots ,\) is unbounded.
$$\begin{aligned} \text{ supp }\ x_n\subset [M,\infty ) (\text{ resp }.\ \text{ supp }\ x_n\subset (0,N]). \end{aligned}$$
$$\begin{aligned} \text{ supp }\ x_n\subset [M,N], n=1, 2, \ldots . \end{aligned}$$
Definition 2.4
If a sequence \((x_n)\) of positive fuzzy numbers is satisfied with FDE (3), then \(x_n\) is said to be a positive solution of FDE (3). \(x\in \Re _F^+\) is called a positive equilibrium of FDE (3) ifLet \(x_n,x\in \Re _F^+, n=0,1,2,\ldots ,\) we say \(x_n\) converges to x as \(n\rightarrow \infty \) if \(\lim _{n\rightarrow \infty }D(x_n,x)=0\).
$$\begin{aligned} x=A+\frac{Bx}{x^2}. \end{aligned}$$
Lemma 2.1
Let \(g: R^+\times R^+\times R^+\times R^+\rightarrow R^+\) be continuous, \(A, B, C, D\in \Re _F^+\). Then
$$\begin{aligned}{}[g(A,B,C,D)]_\alpha =g([A]_\alpha ,[B]_\alpha ,[C]_\alpha ,[D]_\alpha ),\ \ \alpha \in (0,1]. \end{aligned}$$
(8)
Lemma 2.2
[8] Let \(I_x, I_y\) be some intervals of real numbers and let \(f: I_x^2\times I_y^2\rightarrow I_x,\), \(g:I_x^2\times I_y^2\rightarrow I_y\) be continuously differentiable functions. Then for every initial conditions \((x_i,y_i)\in I_x\times I_y (i=-1,0)\), the following system of difference equationshas a unique solution \(\{(x_i,y_i)\}_{n=-1}^{+\infty }\).
$$\begin{aligned} \left\{ \begin{array}{ccc} x_{n+1}=f(x_n,x_{n-1},y_n,y_{n-1}),&{}\\ &{} n=0,1,2,\ldots ,\\ y_{n+1}=g(x_n,x_{n-1},y_n,y_{n-1}).&{} \end{array} \right. \end{aligned}$$
(9)
Definition 2.5
[8] A point \(({\overline{x}},{\overline{y}})\) is called an equilibrium point of system (9) ifThat is, \((x_n,y_n)=({\overline{x}},{\overline{y}})\) for \(n\ge 0\) is the solution of (9), or equivalently, \(({\overline{x}},{\overline{y}})\) is a fixed point of the vector map (f, g).
$$\begin{aligned} {\overline{x}}=f({\overline{x}},{\overline{x}},{\overline{y}},{\overline{y}}),\ \ {\overline{y}}=g({\overline{x}},{\overline{x}},{\overline{y}},{\overline{y}}). \end{aligned}$$
Definition 2.6
[8] Let \(({\overline{x}},{\overline{y}})\) be an equilibrium point of (9). (i) The equilibrium point \(({\overline{x}},{\overline{y}})\) is called locally stable if for every \(\varepsilon >0\), there exists \(\delta >0\) such that for all \((x_i,y_i)\in I_x\times I_y(i=-1,0)\), with \(\mid x_{-1}-{\overline{x}}\mid +\mid x_0-{\overline{x}}\mid<\delta , \mid y_{-1}-{\overline{y}}\mid +\mid y_0-{\overline{y}}\mid <\delta , \) we have \(\mid x_n-{\overline{x}}\mid<\varepsilon , \mid y_n-{\overline{y}}\mid <\varepsilon ,\) for \(n\ge 0.\)
(ii) The equilibrium \(({\overline{x}},{\overline{y}})\) of (9) is called locally asymptotically stable if it is locally stable, and if there exists \(\gamma >0\), such that for all \((x_i,y_i)\in I_x\times I_y(i=-1,0)\), with \(\mid x_{-1}-{\overline{x}}\mid +\mid x_0-{\overline{x}}\mid<\gamma , \mid y_{-1}-{\overline{y}}\mid +\mid y_0-{\overline{y}}\mid <\gamma , \) we have(iii) The equilibrium \(({\overline{x}},{\overline{y}})\) of (9) is called a global attractor if for every \((x_i,y_i)\in I_x\times I_y(i=-1,0)\), we have(iv) The equilibrium \(({\overline{x}},{\overline{y}})\) of (9) is called globally asymptotically stable if it is locally stable and a global attractor.
$$\begin{aligned} \lim _{n\rightarrow \infty }x_n={\overline{x}},\ \ \lim _{n\rightarrow \infty }y_n={\overline{y}}. \end{aligned}$$
$$\begin{aligned} \lim _{n\rightarrow \infty }x_n={\overline{x}},\ \ \lim _{n\rightarrow \infty }y_n={\overline{y}}. \end{aligned}$$
(v) The equilibrium \(({\overline{x}},{\overline{y}})\) of (9) is called unstable if it is not stable.
3 Main results
3.1 Existence of positive fuzzy solution
In this section, we first discuss the existence of the positive fuzzy solution of FDE (3).
Theorem 3.1
Proof
The proof of Theorem is similar to those of Proposition 2.1 [19]. Assume that \((x_n)\) is a sequence of fuzzy numbers and satisfies FDE (3) with initial values \(x_{-1}, x_0.\) Consider the \(\alpha -\)cuts, \(\alpha \in (0,1], n\in N^+\)Applying Lemma 2.1, it follows from (3) and (10) thatUtilizing g-division of fuzzy numbers, one of the following two cases occurs.
$$\begin{aligned} \left\{ \begin{array}{lll} {[}x_n]_\alpha &{}=&{}[L_{n,\alpha },R_{n,\alpha }], \\ &{}&{}\\ {[}A]_\alpha &{}=&{} [A_{l,\alpha }, A_{r,\alpha }], \\ &{}&{}\\ {[}B]_\alpha &{}=&{} [B_{l,\alpha },B_{r,\alpha }]. \end{array} \right. \end{aligned}$$
(10)
$$\begin{aligned}{}[x_{n+1}]_\alpha= & {} [L_{n+1,\alpha },R_{n+1,\alpha }]=\left[ A+\frac{Bx_n}{x_{n-1}^2}\right] _\alpha =[A]_\alpha +\frac{[B]_\alpha \times [x_n]_\alpha }{[x_{n-1}^2]_\alpha }\\= & {} [A_{l,\alpha },A_{r\alpha }]+\frac{[B_{l,\alpha }L_{n,\alpha },B_{r,\alpha }R_{n,\alpha }]}{[L_{n-1,\alpha }^2, R_{n-1,\alpha }^2]}. \end{aligned}$$
Case (i)Case (ii)If Case (i) holds true, from (11), one gets, for \( \alpha \in (0,1], n\in \{0,1,2,\ldots \},\)It is clear that there is a unique solution \((L_{n,\alpha },R_{n,\alpha })\) for any initial value \((L_{k,\alpha },R_{k,\alpha }),k=-1,0,\alpha \in (0,1]\). We only need to show that, for \(\alpha \in (0,1],\ n\in N^+\), \([L_{n,\alpha },R_{n,\alpha }]\) makes certain the positive fuzzy solution \(x_n\) of FDE (3) with the initial condition \(x_{i}, i=0,-1\), andFrom [18], since \(x_j\in \Re _F^+,j=-1,0\), for \(\alpha _i\in (0,1], i=1,2,\) and \(\alpha _1\le \alpha _2\), it hasWe declare that, for \(n=0,1,2,\ldots ,\)By mathematical induction. From (15), for \(n=0,1\), it is true. Suppose that, for \(n\le k, k\in N^+\), (16) holds true, then it follows from (13) and (16) that, for \(n=k+1\),So (16) holds true. Also, from (13), it hasSince \(x_j\in \Re _F^+, j=-1, 0\), and \(A, B\in \Re _F^+\) , then \(L_{i,\alpha },R_{i,\alpha },i=0,-1,\) are left continuous. So, one gets from (17) that \(L_{1,\alpha },R_{1,\alpha }\) are also left continuous. By inductively, we can show that, for \(n\ge 1\), \(L_{n,\alpha }\) and \(R_{n,\alpha }\) are left continuous.
$$\begin{aligned}{}[x_{n+1}]_\alpha =[L_{n+1,\alpha },R_{n+1,\alpha }]=\left[ A_{l,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2}, A_{r,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2}\right] . \end{aligned}$$
(11)
$$\begin{aligned}{}[x_{n+1}]_\alpha =[L_{n+1,\alpha },R_{n+1,\alpha }]=\left[ A_{l,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2}, A_{r,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2}\right] . \end{aligned}$$
(12)
$$\begin{aligned} L_{n+1,\alpha }=A_{l,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2},\ \ R_{n+1,\alpha }=A_{r,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2}. \end{aligned}$$
(13)
$$\begin{aligned}{}[x_n]_\alpha =[L_{n,\alpha },R_{n,\alpha }]. \end{aligned}$$
(14)
$$\begin{aligned} 0<L_{j,\alpha _1}\le L_{j,\alpha _2}\le R_{j,\alpha _2}\le R_{j,\alpha _1},j=-1, 0. \end{aligned}$$
(15)
$$\begin{aligned} L_{n,\alpha _1}\le L_{n,\alpha _2}\le R_{n,\alpha _2}\le R_{n,\alpha _1}. \end{aligned}$$
(16)
$$\begin{aligned} L_{k+1,\alpha _1}= & {} A_{l,\alpha _1}+\frac{B_{l,\alpha _1}L_{k,\alpha _1}}{L_{k-1,\alpha _1}^2} \le A_{l,\alpha _2}+\frac{B_{l,\alpha _2}L_{k,\alpha _2}}{L_{k-1,\alpha _2}^2}=L_{k+1,\alpha _2}\\= & {} A_{l,\alpha _2}+\frac{B_{l,\alpha _2}L_{k,\alpha _2}}{L_{k-1,\alpha _2}^2}\le A_{r,\alpha _2} +\frac{B_{r,\alpha _2}R_{k,\alpha _2}}{R_{k-1,\alpha _2}^2}=R_{k+1,\alpha _2}\\= & {} A_{r,\alpha _2}+\frac{B_{r,\alpha _2}R_{k,\alpha _2}}{R_{k-1,\alpha _2}^2}\le A_{r,\alpha _1}+\frac{B_{r,\alpha _1}R_{k,\alpha _1}}{R_{k-1,\alpha _1}^2}=R_{k+1,\alpha _1} \end{aligned}$$
$$\begin{aligned} L_{1,\alpha }=A_{l,\alpha }+\frac{B_{l,\alpha }L_{0,\alpha }}{L_{-1,\alpha }^2},\ \ R_{1,\alpha }=A_{r,\alpha }+\frac{B_{r,\alpha }R_{0,\alpha }}{R_{-1,\alpha }^2},\ \ \alpha \in (0,1]. \end{aligned}$$
(17)
Secondly, we will prove that \(\text{ supp } x_n=\overline{\bigcup _{\alpha \in (0,1]}[L_{n,\alpha },R_{n,\alpha }]}\) is compact. It need to show that \(\bigcup _{\alpha \in (0,1]}[L_{n,\alpha },R_{n,\alpha }]\) is bounded. Since \(x_j\in \Re _F^+,j=-1,0 \), and \(A,B\in \Re _F^+\), there exist \(M_A>0,N_A>0,N_B>0, M_B>0, M_j>0,N_j>0,j=-1,0,\) such that, \(\forall \alpha \in (0,1]\),Hence from (17) and (18), one can get, for \(n=1\),From which, it follows thatHence, \(\overline{\bigcup _{\alpha \in (0,1]}[L_{1,\alpha },R_{1,\alpha }]}\) is compact, and \(\overline{\bigcup _{\alpha \in (0,1]}[L_{1,\alpha },R_{1,\alpha }]}\subset (0,\infty )\). Deducing inductively, one can get that \(\overline{\bigcup _{\alpha \in (0,1]}[L_{n,\alpha }, R_{n,\alpha }]}\) is compact, moreover, for \(n=1,2,\ldots ,\)And since \(L_{n,\alpha }, R_{n,\alpha }\) are left continuous, from (16) and (21), we get that \([L_{n,\alpha }, R_{n,\alpha }]\) makes certain a positive sequence \(x_n\) satisfying (14).
$$\begin{aligned} \left\{ \begin{array}{l} [A_{l,\alpha },A_{r,\alpha }]\subset [M_A,N_A],\\ \\ {[}B_{l,\alpha },B_{r,\alpha }]\subset [M_B,N_B], \\ \\ {[}L_{j,\alpha },R_{j,\alpha }]\subset [M_j,N_j]. \end{array} \right. \end{aligned}$$
(18)
$$\begin{aligned} {[}L_{1,\alpha },R_{1,\alpha }]\subset \left[ M_A+\frac{M_BM_0}{M_{-1}^2},N_A+\frac{N_BN_0}{N_{-1}^2}\right] ,\alpha \in (0,1]. \end{aligned}$$
(19)
$$\begin{aligned} \bigcup _{\alpha \in (0,1]}[L_{1,\alpha },R_{1,\alpha }]\subset \left[ M_A+\frac{M_BM_0}{M_{-1}^2}, N_A+\frac{N_BN_0}{N_{-1}^2}\right] ,\ \ \alpha \in (0,1]. \end{aligned}$$
(20)
$$\begin{aligned} \overline{\bigcup _{\alpha \in (0,1]}[L_{n,\alpha },R_{n,\alpha }]}\subset (0,\infty ). \end{aligned}$$
(21)
Now we show that, for the initial conditions \(x_{i} (i=0,-1)\), \(x_n\) is the solution of (3). Since, \(\forall \alpha \in (0,1]\),Therefore, \(x_n\) is the solution of FDE (3) with initial conditions \(x_{i}, i=0, -1\).
$$\begin{aligned}{}[x_{n+1}]_\alpha= & {} [L_{n+1,\alpha },R_{n+1,\alpha }]\\= & {} \left[ A_{l,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2}, A_{r,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2}\right] =\left[ A+\frac{Bx_{n}}{x_{n-1}^2}\right] _\alpha . \end{aligned}$$
Suppose that, for the initial values \(x_{i}\in \Re _F^+, i=0,-1\), \({\overline{x}}_n\) is another solution of (3). Then, deducing as above, it has, for \(n\in N^+\)Then, (14) and (22) imply that \([x_n]_\alpha =[{\overline{x}}_n]_\alpha , \alpha \in (0,1],n\in N^+,\) so \(x_n={\overline{x}}_n.\)
$$\begin{aligned}{}[{\overline{x}}_n]_\alpha =[L_{n,\alpha },R_{n,\alpha }],\ \ \ \alpha \in (0,1].\ \ \ \end{aligned}$$
(22)
If Case (ii) occurs, The proof is similar to the proof above. This completes the proof of Theorem 3.1. \(\square \)
3.2 Dynamics of FDE (3)
In order to obtain results on qualitative features of the positive solutions, Case (i) and Case (ii) are considered respectively.
If Case (i) occurs, the following lemma is required.
Lemma 3.1
Consider the following difference equationswhere \(p^2>c>0, y_{i}\in (0,+\infty ), i=0,-1\). Then
$$\begin{aligned} y_{n+1}=p+\frac{cy_n}{y_{n-1}^2},\ \ n=0,1,\ldots , \end{aligned}$$
(23)
$$\begin{aligned} p\le y_n\le \frac{p^3}{p^2-c}+y_3, \ \ n\ge 4. \end{aligned}$$
(24)
Proof
It is clear that, for \(n\ge 1\), \(y_n>p, z_n>q\) from (23). Moreover, for \(n\ge 4\),By induction, it can get that, for \(n-k\ge 3\)Noting \(n-k\ge 3\) is equal to \(k\le n-3\). The proposition is true. \(\square \)
$$\begin{aligned} y_n=p+\frac{cy_{n-1}}{y_{n-2}^2}\le p+\frac{c}{p^2}y_{n-1}, \end{aligned}$$
(25)
$$\begin{aligned} y_n\le & {} p+\frac{c}{p}+\frac{c}{p^4}y_{n-2}\le p+\frac{c}{p}+\frac{c^2}{p^3}+\frac{c^3}{p^6}y_{n-3} \le p+\frac{c}{p}+\frac{c^2}{p^3}+\frac{c^3}{p^5}+\frac{c^4}{p^8}y_{n-4}\nonumber \\\le & {} \cdots \le \sum _{i=1}^{k}\frac{c^{i-1}}{p^{2i-3}}+\frac{c^k}{p^{2k}}y_{n-k} =\frac{p}{1-c/p^2}\left[ 1-\left( \frac{c}{p^2}\right) ^{k}\right] +\frac{c^k}{p^{2k}}y_{n-k}\nonumber \\\le & {} \frac{p^3}{p^2-c}+y_{n-k} \end{aligned}$$
(26)
Theorem 3.2
Consider FDE (3), where \(A, B\in \Re _F^+\), and \(x_{-1},x_0\in \Re _F^+\). Suppose that there exist \(M_A>0, N_A>0, M_B>0, N_B>0\), \(\forall \alpha \in (0,1]\), such thatThen every positive fuzzy solution \(x_n\) of FDE (3) is bounded and persists.
$$\begin{aligned} \left\{ \begin{array}{l} M_A\le A_{l,\alpha }\le A_{r,\alpha }\le N_A, \\ \\ \ M_B\le B_{l,\alpha }\le B_{r,\alpha }\le N_B,\\ \\ M_A^2>M_B,\ \ N_A^2>N_B. \end{array} \right. \end{aligned}$$
(27)
Proof
If Case (i) occurs. Let \(x_n\) be a positive fuzzy solution of FDE (3). It is obvious from (11) that, for \(n=1,2,\ldots ,\ \ \alpha \in (0,1],\)Then, from (21), Lemma 3.1, and \(A_{l,\alpha }\ge M_A\), it has, for \(n\ge 5,\)whereThen since \(x_n\in \Re _F^+\), there exists \(T>0\), satisfying,Therefore, (29) and (30) imply that \([L_{n,\alpha },R_{n,\alpha }]\subset [M_A,T],\) then, for \(n\ge 4, \bigcup _{\alpha \in (0,1]}[L_{n,\alpha },R_{n,\alpha }]\subset [M_A,T]\), so \(\overline{\bigcup _{\alpha \in (0,1]}[L_{n,\alpha },R_{n,\alpha }]}\subseteq [M_A,T]\). Thus the positive fuzzy solution \(x_n\) is bounded and persists.
$$\begin{aligned} M_A\le L_{n,\alpha },\ \ M_A\le R_{n,\alpha }. \end{aligned}$$
(28)
$$\begin{aligned}{}[L_{n,\alpha },R_{n,\alpha }]\subset [M_A, T_\alpha ], \end{aligned}$$
(29)
$$\begin{aligned} T_\alpha = \frac{N_A^3}{N_A^2-N_B}+R_{3,\alpha }. \end{aligned}$$
$$\begin{aligned} T_\alpha \le T,\ \ \ \forall \alpha \in (0,1]. \end{aligned}$$
(30)
To show that the convergence of positive fuzzy solution \(x_n\), we give the following lemmas. \(\square \)
Lemma 3.2
Consider the following difference equationAssume \(p^2>\frac{4c}{3}\). Then the equilibrium \({\overline{y}}\) of (31) is locally asymptotically stable.
$$\begin{aligned} y_{n+1}=p+\frac{cy_n}{y_{n-1}^2},\ n=0,1,2,\ldots , \end{aligned}$$
(31)
Proof
Let \({\overline{y}}\) be an equilibrium of (31), we obtain that \({\overline{y}}=\frac{p+\sqrt{p^2+4c}}{2}.\) The linearized equation of (31) at equilibrium \({\overline{y}}\) isSince \(p^2>\frac{4c}{3}\), it can getBy virtue of Theorem 1.3.7 [8], the equilibrium \({\overline{y}}\) of (31) is locally asymptotically stable. \(\square \)
$$\begin{aligned} y_{n+1}-\frac{2c}{p^2+2c+p\sqrt{p^2+4c}}y_n+\frac{4c}{p^2+2c+p\sqrt{p^2+4c}}y_{n-1}=0, \end{aligned}$$
(32)
$$\begin{aligned} \frac{6c}{p^2+2c+p\sqrt{p^2+4c}}<1. \end{aligned}$$
Lemma 3.3
Proof
From system (23), we obtain a unique equilibrium \({\overline{y}}=\frac{p+\sqrt{p^2+4c}}{2}.\) If \((y_n)\) is a positive solution of (23). LetApplying Lemma 3.1, we get \(0<p<\lambda _1\le \Lambda _1<\infty .\) From (23), it implies thatThis can derive thatThus \(\Lambda _1=\lambda _1\). Namely \(\lim _{n\rightarrow \infty }y_n\) exists. Since \({\overline{y}}\) of (23) is the unique positive equilibrium, then \(\lim _{n\rightarrow \infty }y_n={\overline{y}}\).
$$\begin{aligned} \Lambda _1=\lim _{n\rightarrow \infty }\sup y_n,\ \ \lambda _1=\lim _{n\rightarrow \infty }\inf y_n. \end{aligned}$$
$$\begin{aligned} \Lambda _1\le p+\frac{c\Lambda _1}{\lambda _1^2}, \ \ \lambda _1\ge p+\frac{c\lambda _1}{\Lambda _1^2}. \end{aligned}$$
$$\begin{aligned} \Lambda _1\le \lambda _1. \end{aligned}$$
Combining two lemmas above, we have the following theorem \(\square \)
Theorem 3.3
Theorem 3.4
If \( A_{l,\alpha }^2>\frac{4}{3}B_{l,\alpha }, A_{r,\alpha }^2>\frac{4}{3}B_{r,\alpha }\), for all \(\alpha \in (0,1],\) then every positive fuzzy solution \(x_n\) of FDE (3) converges to the fuzzy positive equilibrium x as \(n\rightarrow \infty .\)
Proof
Suppose that there is a fuzzy number x satisfyingin which \(L_\alpha , R_\alpha \ge 0\). Then, from (33), one getsHence we have from (34) thatLet \(x_n\) be a positive fuzzy solution of FDE (3) satisfying (11). Since \(A_{l,\alpha }^2>\frac{4}{3}B_{l,\alpha }, A_{r,\alpha }^2>\frac{4}{3}B_{r,\alpha }, \alpha \in (0,1]\). Utilizing Lemma 3.3 to system (13), thenThen, from (35), it hasThe proof of the theorem is completed.
$$\begin{aligned} x=A+\frac{Bx}{x^2},\ \ [x]_\alpha =[L_\alpha ,R_\alpha ],\ \ \alpha \in (0,1]. \end{aligned}$$
(33)
$$\begin{aligned} L_\alpha =A_{l,\alpha }+\frac{B_{l,\alpha }L_\alpha }{L_\alpha ^2},\ \ R_\alpha =A_{r,\alpha }+\frac{B_{r,\alpha }R_\alpha }{R_\alpha ^2}. \end{aligned}$$
(34)
$$\begin{aligned} L_\alpha =\frac{A_{l,\alpha }+\sqrt{A_{l,\alpha }^2+4B_{l,\alpha }}}{2},\ \ R_\alpha =\frac{A_{r,\alpha }+\sqrt{A_{r,\alpha }^2+4B_{r,\alpha }}}{2}. \end{aligned}$$
$$\begin{aligned} \lim _{n\rightarrow \infty }L_{n,\alpha }=L_\alpha ,\ \ \lim _{n\rightarrow \infty }R_{n,\alpha }=R_\alpha , \end{aligned}$$
(35)
$$\begin{aligned} \lim _{n\rightarrow \infty }D(x_n,x)=\lim _{n\rightarrow \infty }\sup _{\alpha \in (0,1]}\{\max \{\mid L_{n,\alpha }-L_\alpha \mid , \mid R_{n,\alpha }-R_\alpha \mid \}\}=0. \end{aligned}$$
Secondly, if Case (ii) occurs, i.e., for \(n\in \{0,1,2,\ldots \}, \)The following lemmas are required. \(\square \)
$$\begin{aligned} L_{n+1,\alpha }=A_{l,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2},\ \ R_{n+1,\alpha }=A_{r,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2}, \ \alpha \in (0,1]. \end{aligned}$$
(36)
Lemma 3.4
Consider the following difference equations system.where \(y_{i}, z_{i}\in (0,+\infty ), i=-1,0\). IfThen, for \(n\ge 4\),
$$\begin{aligned} y_{n+1}=p+\frac{dz_n}{z_{n-1}^2},\ \ z_{n+1}=q+\frac{cy_n}{y_{n-1}^2},\ \ n=0,1,\ldots , \end{aligned}$$
(37)
$$\begin{aligned} p>d>1,\ \ q>c>1. \end{aligned}$$
(38)
$$\begin{aligned} p\le y_n\le \frac{p(p^2q^2+pqd)}{p^2q^2-cd}+y_2,\ \ q\le z_n\le \frac{q(p^2q^2+pqc)}{p^2q^2-cd}+z_2. \end{aligned}$$
(39)
Proof
Let \(Y_n=\frac{y_n}{p}, Z_n=\frac{z_n}{q}\), then Eq. (37) can be transformed into the following systemsFrom (40), we have \(Y_n\ge 1, Z_n\ge 1,\) for \(n\ge 1.\) And, for \(n\ge 4\),Deducing inductively, it can conclude that, for \(n-2k\ge 2\),Noting \(n-2k\ge 2\) is equal to \(k\le (n-2)/2\). The proposition is true. \(\square \)
$$\begin{aligned} Y_{n+1}=1+\frac{d}{pq}\frac{Z_n}{Z_{n-1}^2},\ \ Z_{n+1}=1+\frac{c}{pq}\frac{Y_n}{Y_{n-1}^2},\ n=0,1,2,\ldots , \end{aligned}$$
(40)
$$\begin{aligned} \left\{ \begin{array}{l} Y_n\le 1+\frac{d}{pq}Z_{n-1}\le 1+\frac{d}{pq}+\frac{cd}{p^2q^2}Y_{n-2},\\ \\ Z_n\le 1+\frac{c}{pq}Y_{n-1}\le 1+\frac{c}{pq}+\frac{cd}{p^2q^2}Z_{n-2}. \end{array} \right. \end{aligned}$$
(41)
$$\begin{aligned} Y_n\le & {} 1+\frac{d}{pq}+\frac{cd}{p^2q^2}Y_{n-2}\le 1+\frac{d}{pq}+\frac{cd}{p^2q^2}+\frac{cd^2}{p^3q^3}Z_{n-3}\nonumber \\\le & {} 1+\frac{d}{pq}+\frac{cd}{p^2q^2}+\frac{cd^2}{p^3q^3}+\frac{c^2d^2}{p^4q^4}Y_{n-4}\nonumber \\\le & {} \cdots \le 1+\frac{d}{pq}+\frac{cd}{p^2q^2}+\frac{cd^2}{p^3q^3}+\frac{c^2d^2}{p^4q^4}+\cdots +\frac{c^{2k-1}d^{2k}}{p^{2k-1}q^{2k-1}}+\frac{c^{2k}d^{2k}}{p^{2k}q^{2k}}Y_{n-2k}\nonumber \\= & {} \frac{1}{1-\frac{cd}{p^2q^2}}\left[ 1-\left( \frac{cd}{p^2q^2}\right) ^{k-1}\right] +\frac{\frac{d}{pq}}{1-\frac{cd}{p^2q^2}}\left[ 1-\left( \frac{cd}{p^2q^2}\right) ^k\right] +\frac{c^{2k}d^{2k}}{p^{2k}q^{2k}}Y_{n-2k}\nonumber \\\le & {} \frac{p^2q^2+pqd}{p^2q^2-cd}+Y_{n-2k}. \end{aligned}$$
(42)
$$\begin{aligned} Z_n\le & {} 1+\frac{c}{pq}+\frac{cd}{p^2q^2}Z_{n-2}\le 1+\frac{c}{pq}+\frac{cd}{p^2q^2}+\frac{c^2d}{p^3q^3}Y_{n-3}\nonumber \\\le & {} 1+\frac{c}{pq}+\frac{cd}{p^2q^2}+\frac{c^2d}{p^3q^3}+\frac{c^2d^2}{p^4q^4}Z_{n-4}\nonumber \\\le & {} \cdots \le 1+\frac{c}{pq}+\frac{cd}{p^2q^2}+\frac{c^2d}{p^3q^3}+\frac{c^2d^2}{p^4q^4}+\cdots +\frac{c^{2k}d^{2k-1}}{p^{2k-1}q^{2k-1}}+\frac{c^{2k}d^{2k}}{p^{2k}q^{2k}}Z_{n-2k}\nonumber \\= & {} \frac{1}{1-\frac{cd}{p^2q^2}}\left[ 1-\left( \frac{cd}{p^2q^2}\right) ^{k-1}\right] +\frac{\frac{c}{pq}}{1-\frac{cd}{p^2q^2}}\left[ 1-\left( \frac{cd}{p^2q^2}\right) ^k\right] +\frac{c^{2k}d^{2k}}{p^{2k}q^{2k}}Z_{n-2k}\nonumber \\\le & {} \frac{p^2q^2+pqc}{p^2q^2-cd}+Z_{n-2k}. \end{aligned}$$
(43)
Lemma 3.5
Proof
From (37), we obtain a positive equilibrium \(({\overline{y}},{\overline{z}})=\)The linearized equation of (37) at the equilibrium \(({\overline{y}},{\overline{z}})\) ishere \(\Psi _n=(y_n,y_{n-1},z_n,z_{n-1})^T\),Let \(\lambda _i,i=1,2,3,4\) be the eigenvalues of matrix B, \(L=\text{ diag }(l_1,l_2,l_3,l_4)\) be a diagonal matrix, \(l_1=l_3=1,d_i=d_{2+i}=1-i\varepsilon (i=2),\) andClearly, L is an invertible matrix. Calculating \(LBL^{-1}\), one hasFrom \(l_1>l_2>0, l_3>l_4>0\), it implies thatFurthermore, noting (45), we haveIt is clear that B has the same eigenvalues as \(LBL^{-1}\), andTherefore the equilibrium \(({\overline{y}}, {\overline{z}})\) of (37) is locally asymptotically stable. \(\square \)
$$\begin{aligned} \left( \frac{pq-d+c+\sqrt{(pq-d+c)^2+4pqd}}{2p}, \frac{pq-c+d+\sqrt{(pq-c+d)^2+4pqc}}{2q}\right) . \end{aligned}$$
$$\begin{aligned} \Psi _{n+1}=B\Psi _n, \end{aligned}$$
(44)
$$\begin{aligned} B=\left( \begin{array}{cccc} 0&{} 0&{} \frac{d}{{\overline{z}}^2} &{}-\frac{2d}{{\overline{z}}^2}\\ 1&{} 0 &{}0 &{}0 \\ \frac{c}{{\overline{y}}^2}&{} -\frac{2c}{{\overline{y}}^2}&{}0&{} 0\\ 0&{}0&{}1&{}0 \end{array} \right) . \end{aligned}$$
$$\begin{aligned} 0<\varepsilon <\min \left\{ \frac{1}{2}-\frac{3d}{2{\overline{z}}^2}, \frac{1}{2}-\frac{3c}{2{\overline{y}}^2}\right\} . \end{aligned}$$
(45)
$$\begin{aligned} LBL^{-1}= \left( \begin{array}{cccc} 0&{}0&{}\frac{d}{{\overline{z}}^2}l_1l_3^{-1}&{}-\frac{2d}{{\overline{z}}^2}l_1l_4^{-1}\\ l_2l_1^{-1}&{}0&{}0&{}0\\ \frac{c}{{\overline{y}}^2}l_3l_1^{-1}&{}-\frac{2c}{{\overline{y}}^2}l_3l_2^{-1}&{}0&{}0\\ 0&{}0&{}l_4l_3^{-1}&{}0 \end{array} \right) \end{aligned}$$
$$\begin{aligned} l_2l_1^{-1}<1, l_4l_3^{-1}<1. \end{aligned}$$
$$\begin{aligned} \frac{d}{{\overline{z}}^2}l_1l_3^{-1} +\frac{2d}{{\overline{z}}^2}l_1l_4^{-1} =\frac{d}{{\overline{z}}^2} \left( 1+\frac{2}{1-2\varepsilon }\right)<\frac{3d}{{\overline{z}}^2(1-2\varepsilon )}<1, \\ \frac{c}{{\overline{y}}^2}l_3l_1^{-1} +\frac{2c}{{\overline{z}}^2}l_3l_2^{-1} =\frac{c}{{\overline{y}}^2} \left( 1+\frac{2}{1-2\varepsilon }\right)<\frac{3c}{{\overline{y}}^2(1-2\varepsilon )}<1. \end{aligned}$$
$$\begin{aligned} \max _{1\le i\le 4}\mid \lambda _i\mid\le & {} \Vert LBL^{-1}\Vert _\infty \\= & {} \max \left\{ l_2l_1^{-1}, l_4l_3^{-1}, \frac{d}{{\overline{z}}^2}l_1l_3^{-1}+\frac{2d}{{\overline{z}}^2}l_1l_4^{-1}, \frac{c}{{\overline{y}}^2}l_3l_1^{-1}+\frac{2c}{{\overline{y}}^2}l_3l_2^{-1}\right\} <1. \end{aligned}$$
Lemma 3.6
Proof
From (37), we obtain positive equilibrium \(({\overline{y}},{\overline{z}})=\)Suppose that \((y_n,z_n)\) is an arbitrary positive solution of (37). Noting (35)–(37), one haswhere \(h_i, H_i\in (0,+\infty ), i=1,2.\) Then, noting (37) and (46), we haveFrom which we haveWe claim thatSuppose contrarily that \(H_1>h_1\), then from the first inequality of (47), it can conclude \(h_2>H_2\), which is a contradiction. So \(H_1=h_1\). Similarly we can get \(H_2=h_2\). Noting (37) and (48), then \(\lim _{n\rightarrow \infty }y_n={\overline{y}}, \lim _{n\rightarrow \infty }z_n={\overline{z}}.\) The proof of Lemma 3.6 is completed.
$$\begin{aligned} \left( \frac{pq-d+c+\sqrt{(pq-d+c)^2+4pqd}}{2p}, \frac{pq-c+d+\sqrt{(pq-c+d)^2+4pqc}}{2q}\right) . \end{aligned}$$
$$\begin{aligned} \lim _{n\rightarrow \infty }\sup y_n=H_1,\lim _{n\rightarrow \infty }\inf y_n=h_1,\ \ \lim _{n\rightarrow \infty }\sup z_n=H_2,\ \ \ \lim _{n\rightarrow \infty }\inf z_n=h_2. \end{aligned}$$
(46)
$$\begin{aligned} H_1\le p+\frac{dH_2}{h_2^2}, \ \ h_1\ge p+\frac{dh_2}{H_2^2},\ \ \ H_2\le q+\frac{cH_1}{h_1^2},\ \ h_2\ge q+\frac{ch_1}{H_1^2}. \end{aligned}$$
$$\begin{aligned} H_1-h_1\le d\left( \frac{h_2}{H_2^2}-\frac{H_2}{h_2^2}\right) ,\ \ \ H_2-h_2\le c\left( \frac{h_1}{H_1^2}-\frac{H_1}{h_1^2}\right) . \end{aligned}$$
(47)
$$\begin{aligned} H_1=h_1,\ \ H_2=h_2. \end{aligned}$$
(48)
Theorem 3.5
Theorem 3.6
Ifand, for \(n=0,1,2,\ldots ,\)Then every positive fuzzy solution of FDE (3) converges to the positive fuzzy equilibrium x as \(n\rightarrow +\infty .\)
$$\begin{aligned} A_{l,\alpha }>B_{r,\alpha }>1, \ \ A_{r,\alpha }>B_{l,\alpha }>1, \ \forall \alpha \in (0,1]. \end{aligned}$$
(49)
$$\begin{aligned} \frac{R_{n,\alpha }L_{n-1,\alpha }^2}{L_{n,\alpha }R_{n-1,\alpha }^2}\le \frac{B_{l,\alpha }}{B_r,\alpha }, \forall \alpha \in (0,1]. \end{aligned}$$
(50)
Proof
The proof is similar to those of Theorem 3.4. Assume that there is a positive fuzzy number x satisfying (33). From (33), condition (49) and (50), we can getHence we can from (51) have thatSuppose that \(x_n\) is a positive fuzzy solution of FDE (3) such that (10) holds. Noting (50) and (51), utilizing Lemma 3.5 and Lemma 3.6 to (36), we haveThenThe proof of the theorem is completed. \(\square \)
$$\begin{aligned} L_\alpha =A_{l,\alpha }+\frac{B_{r,\alpha } R_\alpha }{R_\alpha ^2},\ \ R_\alpha =A_{r,\alpha }+\frac{B_{l,\alpha }L_\alpha }{L_\alpha ^2}. \end{aligned}$$
(51)
$$\begin{aligned} \left\{ \begin{array}{c} L_\alpha =\frac{A_{l,\alpha }A_{r,\alpha }+B_{r,\alpha }-B_{l,\alpha } +\sqrt{(A_{l,\alpha }A_{r,\alpha }+B_{r,\alpha }-B_{l,\alpha })^2+4A_{l,\alpha }A_{r,\alpha }B_{l,\alpha }}}{2A_{r,\alpha }},\\ \\ R_\alpha =\frac{A_{l,\alpha }A_{r,\alpha }+B_{l,\alpha }-B_{r,\alpha } +\sqrt{(A_{l,\alpha }A_{r,\alpha }+B_{l,\alpha }-B_{r,\alpha })^2+4A_{l,\alpha }A_{r,\alpha }B_{r,\alpha }}}{2A_{l,\alpha }}. \end{array} \right. \end{aligned}$$
(52)
$$\begin{aligned} \lim _{n\rightarrow \infty }L_{n,\alpha }=L_\alpha ,\ \ \lim _{n\rightarrow \infty }R_{n,\alpha }=R_\alpha , \end{aligned}$$
(53)
$$\begin{aligned} \lim _{n\rightarrow \infty }D(x_n,x)=\lim _{n\rightarrow \infty }\sup _{\alpha \in (0,1]}\{\max \{\mid L_{n,\alpha }-L_\alpha \mid , \mid R_{n,\alpha }-R_\alpha \mid \}\}=0. \end{aligned}$$
Remark 3.1
In fuzzy discrete dynamical systems. To find qualitative behavior of solutions for discrete fuzzy difference equation, it is very vital to utilize which operations such as addition, scalar multiplication, division of fuzzy numbers. In [19‐22, 25, 26, 28‐31], Using Zadeh extension principle, the authors obtained dynamical behaviors of some fuzzy difference equations. However, utilizing g-division of fuzzy numbers, Zhang et al. [23, 24] studied the dynamical behaviors of some nonlinear fuzzy difference equations. Compared with the former, The advantage is that the support sets of positive fuzzy solution of latter is smaller than those of former. In fact, it is obvious that the degree of fuzzy uncertainty is reduced by virtue of g-division of fuzzy numbers. Based on this fact above, Therefore, we consider the qualitative behaviors of the fuzzy difference equation with quadratic term by virtue of g-division of fuzzy numbers.
4 Numerical examples
In this section, two illustrative examples are presented to verify the effectiveness of theoretic results.
Example 4.1
Consider the following fuzzy difference equationwhere \(A, B\in \Re _F^+\) and the initial conditions \(x_{i}\in \Re _F^+, i=0,-1\) are as followsFrom (55), one hasFrom (56), one hasTherefore, it follows thatIt is clear that Case (i) occurs, so Eq. (54) can results in a difference equation system with parameter \(\alpha ,\)Therefore, \(A_{l,\alpha }^2>\frac{4}{3}B_{l,\alpha }, A_{r,\alpha }^2>\frac{4}{3}B_{r,\alpha } \forall \alpha \in (0,1],\) and the initial value \(x_{i}\in \Re _F^+,(i=0,-1)\), so applying Theorem 3.2, then every positive solution \(x_n\) of Eq. (54) is bounded and persistent.
$$\begin{aligned} x_{n+1}=A+\frac{Bx_n}{x_{n-1}^2},\ n=0,1,\ldots , \end{aligned}$$
(54)
$$\begin{aligned}&\begin{array}{ll} A(x)=\left\{ \begin{array}{ll} x-3,&{} 3\le x\le 4\\ &{}\\ -x+5,&{} 4\le x\le 5 \end{array} \right. , &{} B(x)=\left\{ \begin{array}{ll} x-4,&{} 4\le x\le 5\\ &{}\\ -x+6,&{} 5\le x\le 6 \end{array} \right. \end{array} \end{aligned}$$
(55)
$$\begin{aligned}&\begin{array}{ll} x_{-1}(x)=\left\{ \begin{array}{ll} 2x-4,&{} 2\le x\le 2.5\\ &{}\\ -2x+6,&{} 2.5\le x\le 3 \end{array} \right. , &{} x_{0}(x)=\left\{ \begin{array}{ll} x-1,&{} 1\le x\le 2\\ &{}\\ -x+3,&{} 2\le x\le 3 \end{array} \right. \end{array} \end{aligned}$$
(56)
$$\begin{aligned}{}[A]_\alpha =\left[ 3+\alpha ,5-\alpha \right] ,\ [B]_\alpha =\left[ 4+\alpha ,6-\alpha \right] ,\ \alpha \in (0,1]. \end{aligned}$$
(57)
$$\begin{aligned}{}[x_{-1}]_\alpha =[2+\frac{1}{2}\alpha ,3-\frac{1}{2}\alpha ],\ \ [x_{0}]_\alpha =[1+\alpha ,3-\alpha ],\ \alpha \in (0,1]. \end{aligned}$$
(58)
$$\begin{aligned} \left\{ \begin{array}{cc} \overline{\bigcup _{\alpha \in (0,1]}[A]_\alpha }=[3,5], &{} \overline{\bigcup _{\alpha \in (0,1]}[B]_\alpha }=[4,6],\\ &{} \\ \overline{\bigcup _{\alpha \in (0,1]}[x_{-1}]_\alpha }=[2,3],&{} \overline{\bigcup _{\alpha \in (0,1]}[x_{0}]_\alpha }=[1,3]. \end{array} \right. \end{aligned}$$
(59)
$$\begin{aligned} L_{n+1,\alpha }=A_{l,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2},\ \ R_{n+1,\alpha }=A_{r,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2},\ \alpha \in (0,1]. \end{aligned}$$
(60)
Example 4.2
Consider Eq. (54), where \(A, B\in \Re _F^+\) and the initial value \(x_{i}\in \Re _F^+, i=0,-1\) are as followsFrom (61), one hasFrom (62), one hasTherefore, it follows thatIt is clear that Case (ii) occurs, so Eq. (54) can result in a difference equation systems with parameter \(\alpha ,\)It is clear that the initial value \(x_{i}\in \Re _F^+(i=0,-1)\), and (47) is satisfied, so applying Theorem 3.6, there is a unique positive equilibrium \({\overline{x}}=(6.3879,7.2749,7.7348)\). Moreover every positive fuzzy solution \(x_n\) of Eq. (54) converges \({\overline{x}}\) with respect to D as \(n\rightarrow \infty .\) (see Figs. 4, 5, 6)
$$\begin{aligned} \begin{array}{ll} A(x)=\left\{ \begin{array}{ll} x-6,&{} 6\le x\le 7\\ &{}\\ -2x+15,&{} 7\le x\le 7.5 \end{array} \right. , &{} B(x)=\left\{ \begin{array}{ll} 2x-3,&{} 1.5\le x\le 2\\ &{}\\ -x+3,&{} 2\le x\le 3 \end{array} \right. \end{array} \end{aligned}$$
(61)
$$\begin{aligned} \begin{array}{ll} x_{-1}(x)=\left\{ \begin{array}{ll} 2x-4,&{} 2\le x\le 2.5\\ &{}\\ -2x+6,&{} 2.5\le x\le 3 \end{array} \right. , &{} x_{0}(x)=\left\{ \begin{array}{ll} x-1,&{} 1\le x\le 2\\ &{}\\ -x+3,&{} 2\le x\le 3 \end{array} \right. \end{array} \end{aligned}$$
(62)
$$\begin{aligned}{}[A]_\alpha =\left[ 6+\alpha ,7.5-\frac{1}{2}\alpha \right] ,\ [B]_\alpha =\left[ 1.5+\frac{1}{2}\alpha ,3-\alpha \right] ,\ \alpha \in (0,1]. \end{aligned}$$
(63)
$$\begin{aligned}{}[x_{-1}]_\alpha =\left[ 2+\frac{1}{2}\alpha ,3-\frac{1}{2}\alpha \right] ,\ \ [x_{0}]_\alpha =[1+\alpha ,3-\alpha ],\ \alpha \in (0,1]. \end{aligned}$$
(64)
$$\begin{aligned}&\overline{\bigcup _{\alpha \in (0,1]}[A]_\alpha }=[6,7.5], \ \overline{\bigcup _{\alpha \in (0,1]}[B]_\alpha }=[1.5,3],\ \overline{\bigcup _{\alpha \in (0,1]}[x_{-1}]_\alpha }=[2,3], \nonumber \\&\overline{\bigcup _{\alpha \in (0,1]}[x_0]_\alpha }=[1,3]. \end{aligned}$$
(65)
$$\begin{aligned} L_{n+1,\alpha }=A_{l,\alpha }+\frac{B_{r,\alpha }R_{n,\alpha }}{R_{n-1,\alpha }^2},\ \ R_{n+1,\alpha }=A_{r,\alpha }+\frac{B_{l,\alpha }L_{n,\alpha }}{L_{n-1,\alpha }^2},\ \alpha \in (0,1]. \end{aligned}$$
(66)
5 Conclusion
In this work, utilizing g-division of fuzzy numbers, the qualitative features of second-order FDE \(x_{n+1}=A+\frac{Bx_{n}}{x_{n-1}^2}\) are studied. The main theoretic results are as follows.
(i) The existence and uniqueness of positive fuzzy solution of FDE (3) are obtained under the positive initial values \(x_{-1},x_0\).
(ii) The positive fuzzy solution is bounded and persistence either Case (i) or Case (ii) occurs. Moreover, every positive fuzzy solution \(x_n\) tend to the unique equilibrium x as \(n\rightarrow \infty \), if \(A_{l,\alpha }^2>\frac{4}{3}B_{l,\alpha }, A_{r,\alpha }^2>\frac{4}{3}B_{r,\alpha }, \alpha \in (0,1]\). And also, every positive fuzzy solution \(x_n\) of FDE (3) converges to the unique equilibrium x as \(n\rightarrow \infty .\) if \(A_{l,\alpha }>B_{r,\alpha }>1, A_{r,\alpha }>B_{1,\alpha }>1\) and \(\frac{R_{n,\alpha }L_{n-1,\alpha }^2}{L_{n,\alpha }R_{n-1,\alpha }^2}\le \frac{B_{l,\alpha }}{B_{r,\alpha }},\alpha \in (0,1], n=0,1,2,\ldots \).
Declarations
Conflict of interest
The authors declare that they have no conflict of interest.
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