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Dynamical behavior of a third-order rational fuzzy difference equation
Advances in Difference Equations volume 2015, Article number: 108 (2015)
Abstract
According to a generalization of division (g-division) of fuzzy numbers, this paper is concerned with the boundedness, persistence and global behavior of a positive fuzzy solution of the third-order rational fuzzy difference equation
where A and initial values \(x_{0}\), \(x_{-1}\), \(x_{-2}\) are positive fuzzy numbers. Moreover, some examples are given to demonstrate the effectiveness of the results obtained.
1 Introduction
It is well known that difference equations appear naturally as discrete analogs and as numerical solutions of differential equations and delay differential equations having many applications in economics, biology, computer science, control engineering, etc. (see, for example, [1–11] and the references therein). Recently there has been a lot of work concerning the global asymptotic stability, the periodicity, and the boundedness of nonlinear difference equations. Moreover, similar results have been derived for systems of two nonlinear difference equations.
Papaschinopoulos and Schinas [12] investigated the global behavior for a system of the following two nonlinear difference equations:
where A is a positive real number, p, q are positive integers, and \(x_{-p},\ldots,x_{0}\), \(y_{-q},\ldots,y_{0}\) are positive real numbers.
In 2005, Yang [13] studied the global behavior of the following system:
where \(p\ge2\), \(q\ge2\), \(r\ge2\), \(s\ge2\), A is a positive constant, and initial values \(x_{1-\max\{p,r\}}, x_{2-\max\{p,r\}},\ldots,x_{0}\), \(y_{1-\max\{q,s\}},y_{2-\max\{q,s\}},\ldots,y_{0}\) are positive real numbers.
In 2012, Zhang et al. [14] investigated the global behavior for a system of the following third-order nonlinear difference equations:
where \(A, B\in(0,\infty)\), and initial values \(x_{-i},y_{-i}\in(0,\infty )\), \(i=0,1,2\).
Ibrahim and Zhang [15] studied dynamics of the third-order system of rational difference equations
\(n=0,1,2,\ldots\) , where the parameters \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), \(\beta _{2}\), \(\gamma_{1}\), \(\gamma_{2}\) and initial conditions \(x_{0}\), \(x_{-1}\), \(x_{-2}\), \(y_{0}\), \(y_{-1}\), \(y_{-2}\) are positive real numbers.
Although difference equations and a system of difference equations are sometimes very simple in their forms, they are extremely difficult to understand through the behavior of their solutions. On the other hand, these models inherently process uncertainty or vagueness. In order to consider these uncertain factors, fuzzy set theory is a powerful tool for modeling uncertainty and for processing vague or subjective information in a mathematical model. Particularly, the use of fuzzy difference equations is a natural way to model the dynamical systems with embedded uncertainty.
Fuzzy difference equation is a difference equation where parameters and initial values are fuzzy numbers, and its solutions are sequences of fuzzy numbers. Due to the applicability of fuzzy difference equation for the analysis of phenomena where imprecision is inherent, this class of difference equations and its applications is a very important topic from theoretical point of view. Recently there has been an increasing interest in the study of fuzzy difference equations (see [16–26]). For example, fuzzy difference equations are suitable in finance problems. Chrysafis et al. [25] studied the fuzzy difference equation of finance. Their research is in finance which is about the alternative methodology to study the time value of money, the method of fuzzy difference equations. Studies have shown that the fuzzy difference equations have a potential to be applied in the theory of fuzzy time series, fuzzy differential equations and stochastic fuzzy differential equations. Readers can refer to [27–32].
Motivated by the discussions above, according to a generalization of division (g-division) of fuzzy numbers, we study the behavior of solutions of the following fuzzy difference equation:
where A and initial conditions \(x_{-i}, y_{-i}\in(0,\infty)\), \(i=0,1,2\), are positive fuzzy numbers.
The aim of this paper is to study the existence of positive solutions of (1). Furthermore, we give some conditions so that every positive solution of (1) is bounded and persistent. Finally, under some conditions we prove that (1) has a unique positive equilibrium x and every positive solution of (1) tends to x as \(n\rightarrow\infty\). Our results extend the result of reference [20].
2 Preliminaries and definitions
For the convenience of the readers, we give the following definitions.
Definition 2.1
[20]
\(A: R\rightarrow[0,1]\) is said to be a fuzzy number if it satisfies conditions (i)-(iv) written below:
-
(i)
A is normal, i.e., there exists \(x\in R\) such that \(A(x)=1\);
-
(ii)
A is fuzzy convex, i.e., for all \(t\in[0,1]\) and \(x_{1}, x_{2}\in R\) such that
$$A\bigl(tx_{1}+(1-t)x_{2}\bigr)\geq\min\bigl\{ A(x_{1}),A(x_{2})\bigr\} ; $$ -
(iii)
A is upper semicontinuous;
-
(iv)
the support of A, \(\operatorname{supp}A=\overline{\bigcup_{\alpha\in(0,1]}[A]_{\alpha }}=\overline{\{x:A(x)>0\}}\) is compact.
For \(\alpha\in(0,1]\), we define the α-cuts of fuzzy number A with \([A]_{\alpha}=\{x\in R: A(x)\geq\alpha\}\) and for \(\alpha=0\), the support of A is defined as \(\operatorname{supp}A=[A]_{0}=\overline{\{x\in R|A(x)>0\}}\). It is clear that \([A]_{\alpha}\) is a closed interval. A fuzzy number is positive if \(\operatorname{supp} A\subset(0,\infty)\).
It is obvious that if A is a positive real number, then A is a fuzzy number such that \([A]_{\alpha}=[A, A]\), \(\alpha\in(0,1]\). Then we say that A is a trivial fuzzy number.
Let A, B be fuzzy numbers with \([A]_{\alpha}=[A_{l,\alpha},A_{r,\alpha}]\), \([B]_{\alpha}=[B_{l,\alpha },B_{r,\alpha}]\), \(\alpha\in[0,1]\), and \(k>0\), we define addition and multiplication as follows:
The collection of all fuzzy numbers with addition and multiplication as defined by Eqs. (2) and (3) is denoted by \(E^{1}\).
Definition 2.2
[20]
The distance between two arbitrary fuzzy numbers A and B is defined as follows:
It is clear that \((E^{1},D)\) is a complete metric space.
Definition 2.3
[33]
Let \(A,B\in E^{1}\) have α-cuts \([A]_{\alpha}=[A_{l,\alpha}, A_{r,\alpha}]\), \([B]_{\alpha}=[B_{l,\alpha}, B_{r,\alpha}]\), with \(0\notin[B]_{\alpha}\), \(\forall\alpha\in[0,1]\). The g-division \(\div_{\mathrm{g}}\) is the operation that calculates the fuzzy number \(C=A\div_{\mathrm{g}}B\) having level cuts \([C]_{\alpha}=[C_{l,\alpha},C_{r,\alpha}]\) (here \([A]_{\alpha}^{-1}=[1/A_{r,\alpha},1/A_{l,\alpha}]\)) defined by
provided that C is a proper fuzzy number (\(C_{l,\alpha}\) is nondecreasing, \(C_{r,\alpha}\) is nondecreasing, \(C_{l,1}\le C_{r,1}\)).
Remark 2.1
According to [33], in this paper the fuzzy number is positive, if \(A\div_{\mathrm{g}} B=C\in E^{1}\) exists, it is easy to see that the following two cases are possible.
Case (i). If \(A_{l,\alpha}B_{r,\alpha}\le A_{r,\alpha }B_{l,\alpha}\), \(\forall\alpha\in[0,1]\), then \(C_{l,\alpha}=\frac {A_{l,\alpha}}{B_{l,\alpha}}\), \(C_{r,\alpha}=\frac{A_{r,\alpha }}{B{r,\alpha}}\).
Case (ii). If \(A_{l,\alpha}B_{r,\alpha}\ge A_{r,\alpha}B_{l,\alpha }\), \(\forall\alpha\in[0,1]\), then \(C_{l,\alpha}=\frac{A_{r,\alpha }}{B_{r,\alpha}}\), \(C_{r,\alpha}=\frac{A_{l,\alpha}}{B{l,\alpha}}\).
The fuzzy analog of the boundedness and persistence (see [18, 19]) is as follows: a sequence of positive fuzzy numbers \((x_{n})\) persists (resp. is bounded) if there exists a positive real number M (resp. N) such that
A sequence of positive fuzzy numbers \((x_{n})\) is bounded and persists if there exist positive real numbers \(M, N>0\) such that
A sequence of positive fuzzy numbers \((x_{n})\), \(n=1, 2, \ldots \) , is unbounded if the norm \(\|x_{n}\|\), \(n=1,2,\ldots\) , is an unbounded sequence.
\(x_{n}\) is a positive solution of (1) if \((x_{n})\) is a sequence of positive fuzzy numbers which satisfies (1). A positive fuzzy number x is called a positive equilibrium of (1) if
Let \((x_{n})\) be a sequence of positive fuzzy numbers and x be a positive fuzzy number, \(x_{n} \rightarrow x\) as \(n\rightarrow\infty\) if \(\lim_{n\rightarrow\infty}D(x_{n},x)=0\).
3 Main results
3.1 Existence of solution of Eq. (1)
Firstly we study the existence of positive solutions of (1). We need the following lemma.
Lemma 3.1
[20]
Let \(f: R^{+}\times R^{+}\times R^{+}\times R^{+}\rightarrow R^{+}\) be continuous, A, B, C, D be fuzzy numbers. Then
Theorem 3.1
Consider Eq. (1) where A is a positive fuzzy number. Then, for any positive fuzzy numbers \(x_{-2}\), \(x_{-1}\), \(x_{0}\), there exists a unique positive solution \(x_{n}\) of (1) with initial conditions \(x_{-2}\), \(x_{-1}\), \(x_{0}\).
Proof
The proof is similar to that of Proposition 2.1 in [19]. Suppose that there exists a sequence of fuzzy numbers \((x_{n})\) satisfying (1) with initial conditions \(x_{-2}\), \(x_{-1}\), \(x_{0}\). Consider the α-cuts, \(\alpha\in(0,1]\),
It follows from (1), (7) and Lemma 3.1 that
Noting Remark 2.1, one of the following two cases holds.
Case (i)
Case (ii)
If Case (i) holds true, it follows that for \(n\in\{0,1,2,\ldots\}\), \(\alpha\in(0,1]\),
Then it is obvious that for any initial condition \((L_{j,\alpha},R_{j,\alpha})\), \(j=-2, -1,0\), \(\alpha\in(0,1]\), there exists a unique solution \((L_{n,\alpha},R_{n,\alpha})\). Now we prove that \([L_{n,\alpha},R_{n,\alpha}]\), \(\alpha\in(0,1]\), where \((L_{n,\alpha},R_{n,\alpha})\) is the solution of system (10) with initial conditions \((L_{j,\alpha},R_{j,\alpha})\), \(j=-2, -1,0\), determines the solution \(x_{n}\) of (1) with initial conditions \(x_{-2}\), \(x_{-1}\), \(x_{0}\) such that
From reference [18] and since \(x_{j}\), \(j=-2, -1,0\), are positive fuzzy numbers for any \(\alpha_{1},\alpha_{2}\in(0,1]\), \(\alpha_{1}\leq\alpha_{2}\), we have
We claim that
We prove it by induction. It is obvious from (12) that (13) holds true for \(n=0,1,2\). Suppose that (13) are true for \(n\le k\), \(k\in \{1,2,\ldots\}\). Then, from (10), (12) and (13) for \(n\le k\), it follows that
Therefore (13) are satisfied. Moreover, from (10) we have
Since \(x_{j}\), \(j=-2, -1, 0\), are positive fuzzy numbers and A is a positive fuzzy number, then we have that \(L_{0,\alpha}\), \(R_{0,\alpha}\), \(L_{-1,\alpha}\), \(R_{-1,\alpha}\), \(L_{-2,\alpha }\), \(R_{-2,\alpha}\) are left continuous. So from (14) we have that \(L_{1,\alpha}\), \(R_{1,\alpha}\) are also left continuous. Inductively we can get that \(L_{n,\alpha}\), \(R_{n,\alpha}\), \(n=1,2,\ldots \) , are left continuous.
Now we prove that the support of \(x_{n}\), \(\operatorname{supp} x_{n}=\overline{\bigcup_{\alpha\in(0,1]}[L_{n,\alpha },R_{n,\alpha}]}\) is compact. It is sufficient to prove that \(\bigcup_{\alpha\in(0,1]}[L_{n,\alpha },R_{n,\alpha}]\) is bounded. Let \(n=1\), since \(x_{j}\), \(j=-2, -1,0 \), are positive fuzzy numbers and A is a positive fuzzy number, there exist constants \(P>0\), \(Q>0\), \(M_{j}>0\), \(N_{j}>0\), \(j=-2,-1,0\), such that for all \(\alpha\in(0,1]\),
Hence from (14) and (15) we can easily get
from which it is obvious that
Therefore (17) implies that \(\overline{\bigcup_{\alpha\in(0,1]}[L_{1,\alpha},R_{1,\alpha}]}\) is compact and \(\overline{\bigcup_{\alpha\in(0,1]}[L_{1,\alpha},R_{1,\alpha}]}\subset (0,\infty)\). Deducing inductively we can easily obtain that \(\overline{\bigcup_{\alpha\in(0,1]}[L_{n,\alpha}, R_{n,\alpha}]}\) is compact, and
Therefore, (13), (18) and since \(L_{n,\alpha}\), \(R_{n,\alpha}\) are left continuous, we have that \([L_{n,\alpha},R_{n,\alpha}]\) determines a sequence of positive fuzzy numbers \(x_{n}\) such that (11) holds.
We prove now that \(x_{n}\) is the solution of (1) with initial conditions \(x_{-1}\), \(x_{0}\). Since for all \(\alpha\in(0,1]\),
we have that \(x_{n}\) is the solution of (1) with initial conditions \(x_{-2}\), \(x_{-1}\), \(x_{0}\).
Suppose that there exists another solution \(\overline{x}_{n}\) of (1) with initial conditions \(x_{-2}\), \(x_{-1}\), \(x_{0}\). Then from arguing as above we can easily prove that
Then from (11) and (19) we have \([x_{n}]_{\alpha}=[\overline{x}_{n}]_{\alpha}\), \(\alpha\in(0,1]\), \(n=0,1,2,\ldots\) , from which it follows that \(x_{n}=\overline{x}_{n}\), \(n=0,1,\ldots\) .
If Case (ii) holds, the proof is similar to that of Case (i). Thus the proof of Theorem 3.1 is completed. □
3.2 Dynamics of Eq. (1)
To study the dynamical behavior of positive solutions of (1), according to Definition 2.3, we consider the following two cases.
Case (i)
Case (ii)
Firstly, if Case (i) holds true, we give the following lemma.
Lemma 3.2
Consider the system of difference equations
where \(p,q\in(1,+\infty)\), \(y_{-2}, y_{-1},y_{0}, z_{-2}, z_{-1},z_{0}\in (0,+\infty)\). Then, for \(n\ge4\),
Proof
From (20) it is clear that \(y_{n}>p\), \(z_{n}>q\) for \(n\ge1\). In view of (20), we obtain for \(n\ge4\) that
Working inductively, we conclude for \(n-k\ge3\) that
Notice that \(n-k\ge3\) is equivalent to \(k\le n-3\). The assertion is true. □
Theorem 3.2
Consider fuzzy difference equation (1), where A is a positive fuzzy number and the initial values \(x_{-1}\), \(x_{0}\) are positive fuzzy numbers. Suppose that there exist positive numbers P, Q for all \(\alpha\in (0,1]\) such that \(1< P\le A_{l,\alpha}\le A_{r,\alpha}\le Q\), then every positive solution \(x_{n}\) of (1) is bounded and persists.
Proof
(i) Let \(x_{n}\) be a positive solution of (1) such that (9) holds. From (8) it is obvious that
Then from \(A_{l,\alpha}\ge P>1\), (25) and Lemma 3.2 we get
where
Then since \(x_{n}\) is a positive fuzzy number, there exists a constant \(T>0\) such that for all \(\alpha\in(0,1]\),
Therefore (25) and (26) imply that \([L_{n,\alpha},R_{n,\alpha}]\subset [P,T]\), \(n\ge4\), from which we get for \(n\ge4\), \(\bigcup_{\alpha\in(0,1]}[L_{n,\alpha},R_{n,\alpha}]\subset[P,T]\), and so \(\overline{\bigcup_{\alpha\in(0,1]}[L_{n,\alpha},R_{n,\alpha }]}\subseteq[P,T]\). Thus the positive solution is bounded and persists. □
To show that every positive solution \(x_{n}\) of system (1) tends to the positive equilibrium x as \(n\rightarrow\infty\), we need the following lemmas.
Lemma 3.3
Consider the difference equation
Assume \(p>\frac{2}{\sqrt{3}}\). Then the equilibrium point of (28) is asymptotically stable.
Proof
Let \(\overline{y}\) be an equilibrium point of (28), it is easy to get \(\overline{y}=\frac{p+\sqrt{p^{2}+4}}{2}\). The linearized equation associated with (28) about equilibrium point \(\overline{y}\) is
Since \(p>\frac{2}{\sqrt{3}}\), we can get
By virtue of Theorem 1.3.7 in [7], the equilibrium point of (28) is asymptotically stable. □
Lemma 3.4
Consider the system of difference equations (20), and assume that \(q>p>\frac{2}{\sqrt{3}}\). Then every positive solution of (20) converges to equilibrium \((\overline{y},\overline{z})= (\frac{p+\sqrt{p^{2}+4}}{2}, \frac {q+\sqrt{q^{2}+4}}{2} )\).
Proof
It is clear that system (20) has a unique equilibrium \((\overline{y},\overline{z})= (\frac{p+\sqrt{p^{2}+4}}{2}, \frac {q+\sqrt{q^{2}+4}}{2} )\). Let \(\{y_{n},z_{n}\}\) be an arbitrary positive solution of (20). Let
From Lemma 3.2, we have \(0< p<\lambda_{1}\le\Lambda_{1}<\infty\), \(0< q<\lambda _{2}\le\Lambda_{2}<\infty\). This and (20) imply that
which can lead to
Thus we have \(\Lambda_{1}=\lambda_{1}\) and \(\Lambda_{2}=\lambda_{2}\). Then \(\lim_{n\rightarrow\infty}y_{n}\) and \(\lim_{n\rightarrow\infty}z_{n}\) exist. From the uniqueness of the positive equilibrium \((\overline{y}, \overline{z})\) of (20), we conclude that \(\lim_{n\rightarrow\infty }y_{n}=\overline{y}\), \(\lim_{n\rightarrow\infty}z_{n}=\overline{z}\). □
Theorem 3.3
Suppose that for all \(\alpha\in(0,1]\), \(A_{l,\alpha}>2/\sqrt{3}\). Then every positive solution \(x_{n}\) of (1) tends to the positive equilibrium x as \(n\rightarrow\infty\).
Proof
Suppose that there exists a fuzzy number x such that
where \(L_{\alpha}, R_{\alpha}\ge0\). Then from (30) we can prove that
Hence from (31) we can have that
Let \(x_{n}\) be a positive solution of (1) such that (7) holds. Since \(A_{l,\alpha}>2/\sqrt{3}\), \(\alpha\in(0,1]\), we can apply Lemma 3.4 to system (10), and so we have
Therefore from (31) we have
This completes the proof of the theorem. □
Secondly, if Case (ii) holds true, it follows that for \(n\in\{ 0,1,2,\ldots\}\), \(\alpha\in(0,1]\),
We need the following lemmas.
Lemma 3.5
Consider the system of difference equations
where \(p,q\in(1,+\infty)\), \(y_{-2}, y_{-1},y_{0}, z_{-2}, z_{-1},z_{0}\in (0,+\infty)\). Then, for \(n\ge4\),
Proof
From (34) it is clear that \(y_{n}\ge p\), \(z_{n}\ge q\) for \(n\ge1\). And for \(n\ge4\) we obtain that
Working inductively, for \(n-2k\ge2\), it can concluded that
Notice that \(n-2k\ge2\) is equivalent to \(k\le(n-2)/2\). The assertion is true. □
Lemma 3.6
Consider the system of difference equations (34), if
are satisfied, then the unique positive equilibrium point \((\overline{y},\overline {z})\) is locally asymptotically stable.
Proof
From (34) the system of difference equations has a unique positive equilibrium point \((\overline{y},\overline{z})= (\frac{pq+\sqrt{p^{2}q^{2}+4pq}}{2p}, \frac{pq+\sqrt{p^{2}q^{2}+4pq}}{2q} )\). The linearized equation of system (33) about the equilibrium point \((\overline{y},\overline{z})\) is
where \(\Psi_{n}=(y_{n},y_{n-1},y_{n-2},z_{n},z_{n-1},z_{n-2})^{T}\), and
Let \(\lambda_{1},\lambda_{2},\ldots,\lambda_{6}\) denote the eigenvalues of matrix B, let \(D=\operatorname{diag}(d_{1},d_{2},\ldots,d_{6})\) be a diagonal matrix, where \(d_{1}=d_{4}=1\), \(d_{i}=d_{3+i}=1-i\varepsilon\) (\(i=2,3\)), and
Clearly, D is invertible. Computing matrix \(DBD^{-1}\), we obtain that
From \(d_{1}>d_{2}>d_{3}>0\), \(d_{4}>d_{5}>d_{6}>0\) it follows that
Furthermore, noting (41) we have
It is well known that B has the same eigenvalues as \(DBD^{-1}\), we have that
This implies that the equilibrium \((\overline{y}, \overline{z})\) of (34) is locally asymptotically stable. □
Lemma 3.7
Consider the system of difference equations (34) if \(p,q\in(1,+\infty)\) and \(\sqrt{3}pq>\max\{3p-q,3q-p\}\). Then every positive solution of (34) converges to the equilibrium point \((\overline{y},\overline{z})\).
Proof
It is clear that system (34) has a unique positive equilibrium point
Let \(\{y_{n},z_{n}\}\) be an arbitrary positive solution of (33). From (33)-(35) we have
where \(l_{i}, L_{i}\in(0,+\infty)\), \(i=1,2\). Then from (34) and (42) we get
from which we have
We claim that
Suppose on the contrary that \(L_{2}>l_{2}\), then from the first inequality of (43) we have \(L_{1}l_{2}< l_{1}L_{2}\), and so \(L_{1}< l_{1}\), which is a contradiction. So \(L_{2}=l_{2}\). Similarly we can prove that \(L_{1}=l_{1}\). Hence from (34) and (43) there exist \(\lim_{n\rightarrow\infty }y_{n}=\overline{y}\), \(\lim_{n\rightarrow\infty}z_{n}=\overline{z}\). This completes the proof of Lemma 3.7. □
Combining Lemma 3.6 with Lemma 3.7, we obtain the following theorem.
Theorem 3.4
Consider the system of difference equations (34). If relations (39) are satisfied, then the unique positive equilibrium \((\overline {y},\overline{z})\) is globally asymptotically stable.
Theorem 3.5
Suppose that
Then every positive solution of (1) tends to the positive equilibrium x as \(n\rightarrow+\infty\).
Proof
The proof is similar to that of Theorem 3.3. Suppose that there exists a fuzzy number x satisfying (30). Then from (30) we can get
Hence we have from (46) that
Let \(x_{n}\) be a positive solution of (1) such that (7) holds. Since (45) is satisfied, we can apply Lemma 3.6 and Lemma 3.7 to system (33), and so we have
Therefore from (46) we have
This completes the proof of the theorem. □
4 Numerical example
Example 4.1
Consider the following fuzzy difference equation:
we take A and the initial values \(x_{-2}\), \(x_{-1}\), \(x_{0}\) such that
From (49) we get
From (50) we get
Therefore, it follows that
From (48), it results in a coupled system of difference equations with parameter α,
Therefore, \(A_{l,\alpha}>2/\sqrt{3}\), \(\forall\alpha\in(0,1]\), and the initial values \(x_{-i}\) (\(i=0,1,2\)) are positive fuzzy numbers. So from Theorem 3.2 we have that every positive solution \(x_{n}\) of Eq. (48) is bounded and persists. In addition, from Theorem 3.3, Eq. (48) has a unique positive equilibrium \(\overline{x}=(2.4142,2.8508,3.3028)\). Moreover, every positive solution \(x_{n}\) of Eq. (48) converges to the unique equilibrium \(\overline{x}\) with respect to D as \(n\rightarrow\infty\) (see Figures 1-4).
Dynamics of system ( 54 ).
The solution of system ( 54 ) in \(\pmb{\alpha=0}\) .
The solution of system ( 54 ) in \(\pmb{\alpha=0.5}\) .
The solution of system ( 54 ) in \(\pmb{\alpha=1}\) .
Example 4.2
Consider the following fuzzy difference equation:
where A and the initial values \(x_{-2}\), \(x_{-1}\), \(x_{0}\) are satisfied
From (56) we get
From (57) we get
Therefore, it follows that
From (55) it results in a coupled system of difference equations with parameter α,
It is clear that (45) is satisfied and the initial values \(x_{-i}\) (\(i=0,1,2\)) are positive fuzzy numbers, so from Theorem 3.5, Eq. (55) has a unique positive equilibrium \(\overline{x}=(1.7808,2.4142, 3.5616)\). Moreover, every positive solution \(x_{n}\) of Eq. (55) converges to the unique equilibrium \(\overline{x}\) with respect to D as \(n\rightarrow\infty\) (see Figures 5-8).
Dynamics of system ( 61 ).
The solution of system ( 61 ) in \(\pmb{\alpha=0}\) .
The solution of system ( 61 ) in \(\pmb{\alpha=0.5}\) .
The solution of system ( 61 ) in \(\pmb{\alpha=1}\) .
5 Conclusion
In this work, according to a generalization of division (g-division) of fuzzy numbers, we study the fuzzy difference equation \(x_{n+1}=A+\frac{x_{n}}{x_{n-1}x_{n-2}}\). The existence of positive solution to (1) is investigated. Furthermore, we obtain the following results:
-
(i)
The positive solution is bounded and persists if \(A_{l,\alpha }>1\), \(\alpha\in(0,1]\), every solution \(x_{n}\) tends to the unique equilibrium x under condition \(A_{l,\alpha}>2/\sqrt{3}\), \(\alpha\in (0,1]\) as \(n\rightarrow\infty\).
-
(ii)
If \(A_{l,\alpha}>1\) and \(\sqrt{3}A_{l,\alpha}A_{r,\alpha }>3A_{r,\alpha}-A_{l,\alpha}\), \(\alpha\in(0,1]\), every solution \(x_{n}\) of (1) converges to the unique equilibrium x as \(n\rightarrow\infty\).
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Acknowledgements
The authors would like to thank the editor and anonymous reviewers for their helpful comments and valuable suggestions, which have greatly improved the quality of this paper. This work was financially supported by the National Natural Science Foundation of China (Grant No. 11361012), the China Postdoctoral Science Foundation (No. 2013T60934), and the Scientific Research Foundation of Guizhou Provincial Science and Technology Department ([2013]J2083).
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The authors indicated in parentheses made substantial contributions to the following tasks of research: drafting the manuscript (QHZ, JZL); participating in the design of the study (ZGL); writing and revision of paper (QHZ). All authors read and approved the final manuscript.
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Zhang, Q., Liu, J. & Luo, Z. Dynamical behavior of a third-order rational fuzzy difference equation. Adv Differ Equ 2015, 108 (2015). https://doi.org/10.1186/s13662-015-0438-2
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DOI: https://doi.org/10.1186/s13662-015-0438-2
MSC
- 39A10
Keywords
- fuzzy difference equation
- boundedness
- persistence
- global asymptotic behavior