Abstract
The purpose of this paper is to give the conditions for the existence and uniqueness of positive solutions and the asymptotic stability of equilibrium points for the following high-order fuzzy difference equation: where is the sequence of positive fuzzy numbers and the parameters and initial conditions are positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.
1. Introduction
It is well known that difference equations are one of the most widely used equations in various subject areas. In particular, many phenomena in the real world often rely on the establishment and solution of mathematical models describing them. In the last twenty five years of the twentieth century, the study on the difference equation developed rapidly and occupied important positions in many applied analyses, such as demographic analysis, finance, and biology (see, e.g., [1–5] and the references therein). However, as the complexity of the modeling systems is increasing, the uncertainty factors in the model increases dramatically. Obviously, the fact that the ordinary difference equation model is used to describe many practical problems is incomplete. In view of this, a fuzzy difference system is a powerful tool to solve the mathematical model with fuzzy parameters and initial values in the real world, and it is an interesting and far-reaching research for modeling uncertainty and for processing vague or subjective information in a mathematical model.
In 1998, DeVault et al. [6] discussed the periodic character of solutions of the following nonlinear difference equation:where
Papaschinopoulos and Schinas [7] researched the following difference equations:where are the positive numbers and are the positive integers.
In 2014, He et al. [8] studied the periodicity of the positive solutions for the following fuzzy max-difference equation:where is a periodic sequence of fuzzy numbers, and initial values are the positive fuzzy numbers with
Zhang et al. [9] investigated the boundedness, persistence, and asymptotic behavior of a positive fuzzy solution of the following third-order fuzzy difference equation using a generalization of division for fuzzy numbers:where are the positive fuzzy numbers.
These bright spots in the above papers deserve our learning. Moreover, in 2017, Khastan [10] discussed the following fuzzy difference equation:where is a sequence of the positive fuzzy numbers and are the positive fuzzy numbers. In this paper, the generalization of division for the fuzzy number is used to investigate the existence, uniqueness, and global behavior of the solution.
Wang and Wang [11] analyzed the following nonlinear difference equation:where the initial conditions are the positive real numbers, are the nonnegative integers, and are the positive constants. Further, in 2017, the author in [12] investigated the asymptotic behavior of the equilibrium points for the following fuzzy difference equation:where is a sequence of the positive fuzzy numbers and the parameters and initial conditions are the positive fuzzy numbers. Besides, some interesting results can be found in [13–18] and the references therein.
Based on the above valuable theoretical results, this paper studies the following high-order fuzzy difference equation:where is a sequence of the positive fuzzy numbers and the parameters and initial conditions are the positive fuzzy numbers. The purpose of this paper is to study the asymptotic behavior of the equilibrium point of the fuzzy difference equation. The main method is to convert the fuzzy difference equation into a rational difference equation according to the fuzzy number theory, and then the properties of the solutions of the fuzzy difference equations are obtained by studying the corresponding constant difference equations. In addition, the theoretical results are verified by numerical examples.
Remark 1. It is well known that difference equations look simple in form, but the properties of their solutions are very complex, especially the dependence on parameters and initial values. The main contribution and innovation of this paper are as follows: (1) based on the practical application, fuzzy parameters and initial values are introduced to the known models, and the new model can better describe the objective natural phenomenon. Obviously, the model in reference [12] is a special case of the model in this paper. (2) To study the new model, some new analytical methods and techniques that is different from those mentioned in the references are obtained. (3) In this paper, the research contents are more rich than the related references. Firstly, the existence and uniqueness of positive fuzzy solutions are proved. Secondly, the nonzero equilibrium points of the corresponding ordinary difference equations which are unstable are obtained by using the linearization method. Finally, it was found that the zero trivial solution of the fuzzy difference equation (8) is asymptotically stable when the parameters of the system are positive trivial fuzzy numbers. (4) The sufficient conditions obtained herein are new, general, and easily verifiable, which provide flexibility for the application and analysis of the high-order fuzzy difference equation.
2. Preliminaries and Notations
For the convenience of readers, the definitions and preliminary results related to the theoretical proof of the paper are given, see [19–23].
Definition 1. For a set , is denoted as the closure of and a function is a fuzzy number if the following conclusions are true:(i) is normal, i.e., there exists such that (ii) is a fuzzy convex set, i.e.,(iii) is upper semicontinuous on ;(iv)The support of , i.e., supp , is compact,where the of are closed intervals, defined as and if supp , then the fuzzy number is obviously positive.
Definition 2. Let be the fuzzy numbers which satisfy and , then the following metric is denoted:where sup is taken for all . Then, is a complete metric space. For the convenience of application in the future, is defined asThus, .
Definition 3. Persistence (resp., boundedness) of fuzzy numbers is defined if there exists a positive real number such that the following conclusions are true: , where is a sequence of the positive fuzzy numbers.
Further, is bounded and persistent if there exist positive real numbers , such that .
Lemma 1. Let be some intervals of real numbers and let , be continuously differentiable functions. Then, for every set of initial conditions , , the following system of difference equations:has a unique solution .
Definition 4. A point is called an equilibrium point of system (12) if that is, for is the solution of difference system (12), or equivalently, is a fixed point of the vector map .
Definition 5. Suppose that is an equilibrium point of system (12), then the following is obtained:(i) is called locally stable if for every there exists such that for any initial conditions with there is s for any (ii) is called an attractor if for any initial conditions (iii) is called asymptotically stable if it is stable and an attractor;(iv) is called unstable if it is not locally stable.
Definition 6. Let be an equilibrium point of the vector map where and are continuously differential functions at . The linearized system of (12) about the equilibrium point is , where is the Jacobian matrix of system (12) about and
Definition 7. Let be four nonnegative integers such that and Split into and into , where denotes a vector with -components of The function possesses a mixed monotone property in subsets of if is monotone nondecreasing in each component of for In particular, if , then it is said to be monotone nondecreasing in
Lemma 2. Assume that , is a system of difference equations and is the equilibrium point of this system, i.e., then the following is obtained:(i)If all eigenvalues of the Jacobian matrix about lie inside the open unit disk , then is locally asymptotically stable;(ii)If one of the eigenvalues of the Jacobian matrix about has a norm greater than one, then is unstable.
Lemma 3. Assume that is a system of difference equations and is the equilibrium point of this system, and the characteristic polynomial of this system about the equilibrium point is , with the real coefficients and . Then, all roots of the polynomial lie inside the open unit disk if and only ifwhere is the principal minor of order of the matrix:
3. Main Results and Their Proofs
The following lemmas are applied to study the existence and uniqueness of a positive solution of the fuzzy difference equation (8).
Lemma 4 (see [24]). Let be a continuous function from into and be the fuzzy numbers, then
Lemma 5 (see [19, 25]). Let us denote such that then and can be seen as functions on which satisfy the following conclusions:(i) is nondecreasing and left continuous;(ii) is nonincreasing and left continuous;(iii).
Conversely, for any functions and defined on which satisfy the above (i)–(iii), there exists a unique such that for any .
Theorem 1. Consider equation (8), where are the positive fuzzy numbers, and then for any positive fuzzy numbers there exists a unique positive solution of (8) with the initial conditions
Proof. Assume that there exists a sequence that is the positive solution of (8) with the positive parameters and initial conditions Consider the From (8) and (16) and Lemma 4, the following is obtained:Hence, for according to the above result, it follows thatAnd then, for any initial conditions it is evident from Lemma 1 that there exists a unique solution of the ordinary difference equations (18).
Conversely, it is proved that the positive solution of equation (8) is determined by with initial conditions and satisfies the following condition:For any , according to Lemma 5, the following is obtained:where are the positive fuzzy numbers.
Next, from the mathematical induction, it is proved that the following conclusion is true, that is,According to (20), (21) is true for Suppose that (21) is true for any , then from (18)–(21), it follows that for ,Hence, (21) is true.
Furthermore, from (18), it follows thatThen, in view of which are the positive fuzzy numbers, it is known that are left continuous from Lemma 5. So, are also left continuous from (23). Moreover, are left continuous using mathematical induction.
Now, it is proven that is supported, i.e., supp is compact. It is easy to know that it is enough to prove that is bounded.
Let , from which are the positive fuzzy numbers, then there exist constants such that for all ,Therefore, relations (23) and (24) imply thatfrom which the following is obtained:From (26), is compact and . Moreover, under the mathematical induction method, it is also proven that is compact andSince (21) and (24) and are left continuous, from Lemma 5, a sequence of positive fuzzy numbers is determined such that (8) holds.
Furthermore, it is proven that is the solution of (6) with initial conditions Since for all ,it is claimed that is the solution of (6) with initial conditions
Now, its uniqueness by contradiction is proven in the following. Assume that there exists another positive solution of (8) with initial conditions and according to a similar discussion above, it holds thatand then, from (19) and (29), one has thatand from the above result, it is obtained that the sequence is a unique solution of (6) with initial conditions so the proof is completed.
In the following result, the asymptotic behavior of the equilibrium point of (6) is further investigated. From the above proof process, it is known that if is the unique positive solution of (6) with the initial values then the following is obtained:Hence, it is known that satisfies the family of system (18). The following corresponding ordinary parametric systems to study the asymptotic behavior of equation (8) are constructed:where the parameters and initial conditions are the positive real constants. Clearly, system (32) has a unique solution for any initial conditions from Lemma 1.
Now, it is easily obtained that system (32) has the following three equilibrium points:if , and system (32) has the fourth positive equilibrium point :Next, the asymptotic behavior of these four equilibrium points in detail is analyzed.
Theorem 2. The equilibrium point is locally asymptotically stable.
Proof. Let be multivariate functions defined bythus, it holds thatMoreover, the linearized system of (32) about the equilibrium point is provided bywhereand the corresponding characteristic polynomial of is as follows:Obviously, all eigenvalues of about lie in an open unit disk ; hence, from Lemma 2, the equilibrium point is locally asymptotically stable, and then the proof ends.
Theorem 3. The equilibrium point is unstable.
Proof. From (36), the linearized systems of (32) about the equilibrium point is as follows:whereand the characteristic polynomial of (40) is given byIt is obvious that there exists so that Thus, a root of the characteristic equation (42) lies outside the unit disk. According to Lemma 2, the equilibrium point of (32) is unstable, and the proof is complete.
Theorem 4. The equilibrium point is unstable.
Proof. From (36), the linearized systems of (36) about the equilibrium point is provided bywhereand the characteristic polynomial of (43) is given bywhich is the same as with (42); therefore, the equilibrium point of (32) is unstable,, and the proof is complete.
Theorem 5. If , then the equilibrium point is unstable.
Proof. From (36), the linearized systems of (32) about the equilibrium point is provided bywhereand the characteristic polynomial of (46) is given byAccording to Lemma 3, the following is obtained:It is obvious that not all From Lemma 2, the equilibrium point is unstable, and then the proof is completed.
Lemma 6 (see [12]). Let be some intervals of real numbers and assume that and be continuously differentiable functions satisfying a mixed monotone property. If there existssuch thatthen there exist and satisfying
Moreover, if , then equation (12) has a unique equilibrium point and every solution of (12) converges to .
Theorem 6. If then the equilibrium point of system (32) is a global attractor for any conditions .
Proof. In view of system (32) can be changed toLet be a function defined bySetwhich means that and have a mixed monotone property.
Letthen,Obviously, since then from system (32) and Lemma 6, there exist such thatThus, Moreover, from then
Hence, it is obtained that the equilibrium point of system (32) is a global attractor from Lemma 6. The proof is completed.
Moreover, from Definition 5, it is obtained that the equilibrium point of system (32) is asymptotically stable.
It is obvious that the fuzzy difference (6) has the trivial solution . Next, the stability of the trivial solution is discussed. To do this, first, the following definition is introduced.
Definition 8. The trivial solution of the fuzzy difference equation (8) is said to be(i)Stable, if given , and there exists with which implies for any , such that for any the solution ;(ii)Attractive if there is such that , and one has(iii)Asymptotically stable if (i) and (ii) hold simultaneously.
Theorem 7. If the parameters are the positive trivial fuzzy numbers, i.e., positive real numbers, and the initial conditions are the positive fuzzy numbers with , then the trivial solution of the fuzzy difference equation (8) is asymptotically stable with respect to as .
Proof. The result follows from Theorem 2 and Theorem 6.
4. Numerical Simulation
Numerical examples are given in this section to confirm the results of the previous sections and to support the theoretical discussion. The example gives the asymptotic behavior of the solution of the fuzzy difference system (8). The following fuzzy difference equation is considered.
Example 1. Let , and the following fuzzy difference equation is considered:where are the positive real numbers. By Theorem 7, In addition, from Theorem 7, the initial conditions with are denoted such thatFrom (61), the corresponding triangular fuzzy numbers are obtained:From (60), the parameters and initial values satisfy the following system of nonlinear difference equation with parameter :It is easy to prove that , for , namely, the conditions of Theorem 7 is satisfied. So from Theorem 7, the trivial solution of the fuzzy difference (6) is asymptotically stable with respect to as (see Figures 1–4).
5. Conclusion
In this paper, a variational iterative method for the nonlinear fuzzy difference equations is proposed. This method is a powerful tool for solving various fuzzy difference equations and can also be applied to other nonlinear differential equations or difference equations in mathematical physics. Numerical simulation results show that this method is effective and simple. Variational iterative method is an effective method to deal with nonlinear structures. MATLAB 2016 software package was used for calculation.
The purpose of this paper was to study the dynamic behavior for a class of nonlinear high-order fuzzy difference equations. Firstly, the existence and uniqueness of positive fuzzy solutions are proved. Secondly, it is obtained that the nonzero equilibrium points of the corresponding ordinary difference equations (32) are unstable by using the linearization method. Finally, it is found that the trivial solution of fuzzy difference (6) is asymptotically stable when the parameters are positive trivial fuzzy numbers. In particular, an illustrate example is given to show the effectiveness of the obtained results. In addition, the sufficient conditions obtained in this paper are very simple and provide some flexibility for the application and analysis of nonlinear fuzzy difference equations.
Data Availability
The data used to support the findings of this work are included within this article.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work was supported by the Scientific Research Fund of the Yunnan Provincial Education Department of China under grant number 2018JS737 and the Program for Innovative Research Team (in Science and Technology) in University of Yunnan Province.