Abstract
We study the qualitative behavior of the positive solutions of a second-order rational fuzzy difference equation with initial conditions being positive fuzzy numbers, and parameters are positive fuzzy numbers. More precisely, we investigate existence of positive solutions, boundedness and persistence, and stability analysis of a second-order fuzzy rational difference equation. Some numerical examples are given to verify our theoretical results.
1. Introduction and Preliminaries
Recently, studying the qualitative behavior of difference equations and systems is a topic of a great interest. Applications of discrete dynamical systems and difference equations have appeared recently in many areas such as ecology, population dynamics, queuing problems, statistical problems, stochastic time series, combinatorial analysis, number theory, geometry, electrical networks, neural networks, quanta in radiation, genetics in biology, economics, psychology, sociology, physics, engineering, economics, probability theory, and resource management. Unfortunately, these are only considered as the discrete analogs of differential equations. It is a well-known fact that difference equations appeared much earlier than differential equations and were instrumental in paving the way for the development of the latter. It is only recently that difference equations have started receiving the attention they deserve. Perhaps this is largely due to the advent of computers where differential equations are solved by using their approximate difference equation formulations. The theory of discrete dynamical systems and difference equations developed greatly during the last twenty-five years of the twentieth century. The theory of difference equations occupies a central position in applicable analysis. There is no doubt that the theory of difference equations will continue to play an important role in mathematics as a whole. Nonlinear difference equations of order greater than one are of paramount importance in applications. It is very interesting to investigate the behavior of solutions of a system of higher-order rational difference equations and to discuss the local asymptotic stability of their equilibrium points. Systems of rational difference equations have been studied by several authors. In particular, there has been a great interest in the study of the attractivity of the solutions of such systems. For some interesting results related to the qualitative behavior of the systems of nonlinear difference equations, we refer the interested reader to [1–4].
It is an interesting mathematical problem to investigate the dynamics of nonlinear fuzzy difference equations. For basic theory of fuzzy difference equations we refer to [5]. For qualitative behavior of fuzzy difference equations, one can see [6–11].
Xiao and Shi [12] studied the qualitative behavior of a rational difference equation of the form where the parameter and initial conditions , are positive real numbers.
Our aim in this paper is to investigate the qualitative behavior of following second-order fuzzy rational difference equation: where is positive fuzzy number and initial conditions , are positive fuzzy numbers.
For further study of (2), we need some basic definitions related to fuzzy difference equations. For more details, we refer to [5–11].
Definition 1. Let be a normed space. One will denote by the family of all fuzzy sets which have the following properties: (i) is normal; that is, there exists an in such that ,(ii) is fuzzy convex; that is, for all , and ,(iii) is upper semicontinuous,(iv)the support of , supp , is compact.
An element is called a fuzzy vector. If , then an element is called fuzzy number.
Remark 2. In particular, if is a family of compact intervals in such that(i) for all ,(ii) whenever is an increasing sequence such that for , then the family represents the -level sets for the fuzzy number , defined by
Moreover, -cuts of are given by for . We say a fuzzy number is positive if supp . Let be positive fuzzy numbers with and for ; then we consider the following metric: A sequence of fuzzy numbers is said to be bounded and persists if there exist positive real numbers such that supp for . For more details of boundedness and persistence of fuzzy difference equations, one can see [7–10]. Moreover, if is a sequence of positive fuzzy numbers and is any positive fuzzy number such that and for and , then converges to with respect to metric , if .
Definition 3. Let be a positive solution of (2) and let be its equilibrium point; that is, ; then(i) is stable, if for given there exists such that if for , then for all ;(ii) is said to be unstable if it is not stable;(iii) is locally asymptotically stable, if it is stable and ;(iv) is said to be an attractor if ;(v) is globally asymptotically stable, if it is an attractor and locally asymptotically stable.
2. Main Results
Arguing as in [8, 11] we have the following theorem for the existence of positive solutions of (2).
Theorem 4. Let be a positive fuzzy number; then (2) has unique positive solution with initial conditions , being positive fuzzy numbers.
Proof. Let be a positive solution of (2). The -cuts of positive fuzzy numbers and are given by Then, from (2), one has Hence, one has the following system:for , and . Let be a solution of system (7) with initial conditions , , , for every . Then, it is sufficient to prove that is a solution of our original system (2) with initial conditions , and for every and . Moreover, for any fuzzy number with , one has that is nondecreasing and left continuous and that is nonincreasing and left continuous for . As , are positive fuzzy numbers, for with , one hasNow, it is easy to show by induction on that Similarly, by induction on , one can show that , are left continuous for all , and . Furthermore, it can be inductively proved that ; that is, is compact. Hence, determines a sequence of positive fuzzy numbers such that for every and .
In order to study the further dynamics of the fuzzy difference equation (2) we will use the results concerning the behavior of the solutions of the corresponding system of two parametric ordinary difference equations given by where the parameters are positive real numbers, and initial conditions , , , are positive real numbers.
Let , ; then and are equilibrium points of system (10). Moreover, is unique positive equilibrium of (10).
Theorem 5. Let be any solution of system (10); then the following statements are true. (1)For every the following results hold: with initial conditions , , , .(2)For the equilibrium point of system (10) the following results hold.(i)Let and ; then equilibrium point of system (10) is locally asymptotically stable.(ii)If or , then equilibrium point of system (10) is unstable.(3)If and , then the unique positive equilibrium point of system (10) is unstable.(4)Let and ; then the equilibrium point of system (10) is globally asymptotically stable.
Proof. (1) It follows from (10) thatfor all . Moreover, consider the following second-order linear difference equations: The solutions of these equations are given byfor all , where , , , and depend on initial conditions , , , . Let , , , ; then by comparison we obtain (2) To construct corresponding linearized form of system (10), we consider the following transformation: where , , , and . The linearized system of (10) about the equilibrium point is given by whereThe characteristic polynomial of is given by The roots of characteristic polynomial (19) are given by (i)Since all eigenvalues of Jacobian matrix about lie in open unit dick if and only if and , the equilibrium point is locally asymptotically stable.(ii)Similarly, all eigenvalues of Jacobian matrix about lie outside open unit dick if and only if and . Hence, the equilibrium point is unstable in this case.
(3) The linearized system of (7) about the equilibrium point is given by wherewhere . The characteristic polynomial of is given by The roots of characteristic polynomial (23) are given by It is sufficient to prove that any one of these roots has absolute value greater than one. For this consider (4) Assume that and ; then from (i) of (2) of Theorem 5 is locally asymptotically stable. Let be any positive solution of (10); then it remains to show that and . Moreover, and if and only if and , respectively. It follows from (1) of Theorem 5 thatHence, .
Next, we discuss the rate of convergence of positive solutions of system (10) which converge to equilibrium point of this system. Similar methods can be found in [1–4]. The following result gives the rate of convergence of solutions of a system of difference equations where is an -dimensional vector, is a constant matrix, and is a matrix function satisfying as , where denotes any matrix norm which is associated with the vector norm
Proposition 6 (Perron’s theorem; see [13]). Suppose that condition (28) holds. If is a solution of (27), then either for all large or exists and is equal to the modulus of one the eigenvalues of matrix .
Proposition 7 (see [13]). Suppose that condition (28) holds. If is a solution of (27), then either for all large or exists and is equal to the modulus of one the eigenvalues of matrix .
Let be any solution of system (10) such that and , where . To find the error terms, one has from system (10) Let , and ; then one has whereMoreover, we obtain Now the limiting system of error terms about can be written as which is similar to linearized system of (10) about the equilibrium point .
Using Proposition 6, one has following result.
Theorem 8. Assume that is a positive solution of system (7) such that and . Then, the error vector of every solution of (10) satisfies both of the following asymptotic relations: where are the characteristic roots of Jacobian matrix .
The following theorem is similar to Theorem of [11].
Theorem 9. Assume that is a positive solution of (2) and ; then is bounded and persists. Moreover, converges to .
Proof. Consider the following system: whereLet be a solution of system (38) with initial conditions , , , , where for are given as Then, it follows thatHence, by induction one has , for all . Assume that ; then it follows that , . Hence, the solution of system (38) is bounded and persists, and so is the solution of (2). Next, from (4) of Theorem 5, it is easy to see that . Then, it follows that
Theorem 10. Assume that is a positive real number and ; then unique positive equilibrium point of (2) is unstable.
Proof. Let be a positive solution of (2) and let be a positive real number with . Then, from (1) of Theorem 5, either and or and . Hence, there are no positive numbers such that . This completes the proof.
3. Examples
Example 1. Consider system (10) with initial conditions , , , . Moreover, choose the parameters . Then, system (10) can be written as with initial conditions , , , . In this case, and . Moreover, for system (43) the plot of is shown in Figure 1 and the plot of is shown in Figure 2.
Example 2. Consider system (10) with initial conditions , , , . Moreover, choose the parameters , . Then, system (10) can be written as with initial conditions , , , . In this case, and . Moreover, for system (44) the plot of is shown in Figure 3 and the plot of is shown in Figure 4.
Example 3. Consider system (10) with initial conditions , , , . Moreover, choose the parameters , . Then, system (10) can be written as with initial conditions , , , . In this case but . Moreover, for system (45) the plot of is shown in Figure 5 and the plot of is shown in Figure 6.
Conflict of Interests
The author declares that he has no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by the Higher Education Commission of Pakistan.