Boundedness and Asymptotic Behaviour of the Solutions for a Third-Order Fuzzy Difference Equation

: Our aim in this paper is to investigate the dynamics of a third-order fuzzy difference equation. By using new iteration method for the more general nonlinear difference equations and inequality skills as well as a comparison theorem for the fuzzy difference equation, some sufficient conditions which guarantee the existence, unstability and global asymptotic stability of the equilibriums for the nonlinear fuzzy system are obtained. Moreover, some numerical solutions of the equation describing the system are given to verify our theoretical results


Introduction
Nonlinear difference equation is an important mathematical models which describe the relationship between the real world phenomenons. It not only enriched the theory of mathematics, but also solved the practical problem, such as the fields of the number of population structure analysis, economic, genetic, biology etc. (see, eg., [1][2][3][4][5] and the references therein). In recent years, research on discrete systems has become a hot problem. (see, eg., [6,7]).
However, the traditional mathematics theories and methods seem to be inadequate in the face of fuzzy phenomenon. Based on this background, the fuzzy difference systems are a powerful tool which can be used to study better some uncertain phenomenons. In recent decades, the fuzzy mathematics theory and its applications have achieved fruitful results. In view of the fact, the fuzzy difference equation system has attracted more and more interest which further enriches the research of the difference system.
In 1998, DeVault et al [8] discussed the existence, boundedness, oscillation behavior of the positive solutions and the global asymptotic behavior of the equilibrium points for the nonlinear difference equation. Besides, Zhang et al [10] researched the following nonlinear fuzzy difference equation. 1 1 , 0, 1, 2, ,  [11] continuously proved similar conclusion for the follow first-order fuzzy difference equation 1 , 0, 1, 2, , More recently, Wang et al [12] investigate the existence and uniqueness of the positive solutions and the asymptotic behavior of the equilibrium points of the following fuzzy difference equation.
Motivated by the discussions above, the purpose of this paper is to discuss the existence and uniqueness of the positive solutions and the asymptotic behavior of the equilibrium points for the following third-order fuzzy difference equation.
x is a sequence of positive fuzzy numbers, the parameters , , A B C and the initial conditions are positive fuzzy numbers. This paper is arranged as follows: in Section 2, some definitions and preliminary results are given. The main results and related proofs are obtained in Section 3. Finally, some numerical examples are used to illustrate our theoretical results.

Preliminaries and Notations
For the convenience of readers, some definitions and preliminary results related to the theoretical proof of this paper are given，see [31][32][33][34][35].
n n n k n n n l n n n n k n n n l x f x x x y y y n y g x x x y y y Has a unique solution { } x y I I ∈ × x y is called asymptotically stable if it is stable, and ( , ) x y is also attractor.
(iv) ( , ) x y is called unstable if it is not locally stable. = … … where f and g are continuously differential functions at ( , ) x y The linearized system of (7) about the equilibrium point ( , ) x σ denotes a vector with σ -components of x . We say that the function (ii) If one of eigenvalues of the Jacobian F J matrix about X has norm greater than one, then X is unstable.

Main Results
The following lemmas are applied to study the existence and uniqueness of the positive solutions of equations (6).
The following results are true from (6), (8) and Lemma 3.1 ] n n n n n n n n n n n Ax Ax [ , ] l n r n l l n n n r r n n n l n n r r n n n r n n l l n n n Then, for any initial conditions , , n n L R α α of the systems (9).
Conversely, it is proved that the positive solution { } n x of the equation (6) is determined by , , For any 1   Next, the following conclusions is proved to be true by mathematical induction.
From (11), the conclusions (12) Thus, we just prove that   (13) and (15) So, from (16) , l n r n r r n n n l l n n n It follows from (10) and (17) n n L R α α satisfies the systems (9). To conveniently study the asymptotic behavior of Eq. (6), according to the systems (9), the corresponding linearized form of the system is constructed as follows Proof. Using the induction method to prove (19). This assertion is true for 0. k = Suppose that (19) The proof is completed. Furthermore, it is easy to know that the systems (18) has two equilibrium points For the two equilibrium points, the following results are shown clearly.
Moreover, the linearized system of Eq. (18) about the equilibrium point 1 X is given by The characteristic equation of the matrix 1 D is shown as Such that for any 0, n > such that for any , 2, 1, 0,

Numerical Simulation
In this section a numerical example is given in order to support our theoretical results. The example represents the asymptotically behavior of solutions for the fuzzy difference system (6).
Example 4.1 Consider the following fuzzy difference equation x In view of (29)

Conclusion
The main purpose of this paper is to deal with the dynamics behavior for a class of nonlinear third-order fuzzy difference equations. Firstly, the existence and uniqueness of positive fuzzy solutions is proved. Secondly, It is obtained that the nonzero equilibrium points of the corresponding ordinary difference equations is unstable by using linearization method. Finally, It is find that the zero trivial solution of the fuzzy equation is stable when the parameters , , A B C are positive real numbers, A B < and the initial conditions are any positive fuzzy numbers. In particular, an example is given to show the effectiveness of the obtained results. In addition, the sufficient conditions obtained in this paper are very simple, which provide flexibility for the application and analysis of nonlinear fuzzy difference equation. For further work, it is our next research target to study the higher order fuzzy difference equations using new iteration method, inequality skills and comparison theorem.