On dynamic behavior of second-order exponential-type fuzzy difference equation
Introduction
Discrete time dynamical systems are usually described by difference equations. The theory of difference equations and their wide applications have attracted scholars' interests. In the past decades, the study of dynamical behaviors of difference equation and system of difference equations has become an important topic in applied mathematics. Particularly, Agarwal published a monograph entitled Difference Equations and Inequalities in 1992 [1]. This book is an in-depth survey of the field hitherto. In 1993, Kocic and Ladas summarized and published a monograph entitled Global Behavior of Nonlinear Difference Equations of Higher Order with Application [14]. The book proposes many open problems, some of which are still unresolved to date. In recent years, many researchers have conducted extensive investigations on dynamical behaviors of nonlinear difference equation, systems of nonlinear difference equations (see, e.g. [8], [9], [10], [13], [15], [16], [23], [29], [32], [37]).
For example, Ozturk et al. [23] studied boundedness, convergent rate and periodic character of the positive solutions for the second-order exponential-type crisp difference equation where and the initial values .
Din et al. [10] discussed the following system of exponential-type difference equations, and obtained useful results including boundedness and persistence of positive solutions, existence and uniqueness of positive equilibrium, local and global behaviors of the system. where and initial values are positive constants.
The identification of the parameters and the initial values of difference equations stems from statistical methods, the choice of some methods adapted to the problem under investigation and data experimentally obtained. These models, even within the classical deterministic approach, incur indeterminacy (fuzzy uncertainty) due to the initial values, the state variables and parameters of the model. In real world, we have learned to dispose of fuzzy uncertainty. Scientists also find that fuzzy uncertainty is a significant and consequential factor in most applications. In 1965, Zadeh [31] introduced fuzzy sets, and thereafter, the theory and application of fuzzy sets have been developed rapidly [2], [7], [20].
A new approach to take uncertainty into account is the fuzzification of crisp difference equations. Fuzzy difference equation is a generalization of crisp difference equation. The parameters or the initial conditions of the model are fuzzy numbers and its solutions are sequences of fuzzy numbers. They play a paramount role analyzing and modeling real world phenomena, e.g. finance problems, biological model [4], [5], [6].
Making a historical flash back for the fuzzy difference equation we study in this paper, we should mention that in 2002, Lakshmikantham and Vatsala [20] proposed the basic theory of fuzzy difference equation. Constructing a Lyapunov-type function and in terms of ordinary difference equations, they obtained a comparison theorem of fuzzy difference equations. Mondal et al. [22] studied a second order linear fuzzy difference equation by Lagrange's multiplier method. Khastan [17], applying the concept of Hukuhara difference (H-difference) for fuzzy numbers, studied fuzzy Logistic difference equation and obtained global behaviors of the solutions for two corresponding equations. Papaschinopoulos and Papadopoulos [24], using Zadeh Extension Principle, studied global behaviors of the fuzzy difference equation , where A and B are positive fuzzy numbers. Additionally, they [25] investigated the fuzzy difference equation , where A is a positive fuzzy number, and . Subsequently, using the same approach, many researchers studied different fuzzy difference equations and have come to many conclusions. We refer the reader to [11], [21], [26], [28], [30], [33], [35], [36].
On the other hand, in set-valued and fuzzy analysis, the standard Minkowski addition and multiplication are not often invertible. However, the inversion of addition and multiplication is basic requirements in interval and fuzzy analysis, with applications to the solution of equations, interval and fuzzy differential equations. To overcome these limitations, Stefanini [27] suggested a generalization of division for fuzzy numbers called g-division and put forward a new method to study some fuzzy linear equations.
It should be noted that the main advantages of g-division, is the reduction of the imprecision of fuzzy solutions because of the decrease of the length of the supports. In [33], Zhang et al. applied g-division of fuzzy numbers to study the global behaviors of a third order rational fuzzy difference equation. Khastan and Alijani [18] studied the same model as [24]. They obtained two different solutions and diameters of solutions are smaller than those of solution obtained by Zadeh Extension Principle. Recently, many researchers investigate global behaviors of fuzzy difference equations by virtue of g-division [19], [34].
In fact, there is no uniform or the most important method so far to study the fuzzy difference equations. The basic way to account for uncertainty is the fuzzification of crisp difference equations. In these approaches, in terms of the different formulations of the unique crisp difference equation, one may bring on different fuzzy difference equations, in each of which, depending on different interpretations of fuzzy operators, e.g. division of fuzzy numbers, one may obtain several solutions with different behaviors. These situations are usually regarded as unnatural from a certain standpoint, since, in any model, one hopes the solution to depict accurately the real behavior of the system. However, these facts are boiled down to an advantage in [3], [18], because of the existence of different choices which can be examined through the scrutiny of the physical features of the particular phenomena.
Motivated by the above facts, in this work, using g-division, we provide a quantitative study of a second-order exponential-type fuzzy difference equation where and the initial values are positive fuzzy numbers.
The major contribution of this paper is to study the existence, uniqueness and global behavior of the solution to Eq. (1.1) by virtue of g-division of fuzzy numbers [27]. Depending on the nature of problem, we may choose an appropriate formulation which describes better the behaviors of the modeled real world. Similarly to the results of [19], we obtain two different results and do not give an answer to the question about which is the best formulation for the problem or the closest to the classical case. In a sense, the method and results can be potentially applied to study some discrete time dynamic models with fuzzy uncertainty.
The structure of the paper is as follows. Section 2 introduces some preliminaries and known results that are used in the sequel. According to g-division, the existence and global behavior of fuzzy difference equation (1.1) are obtained in Section 3. In Section 4, two examples are presented to illustrate the validity of our results. Finally, conclusion is drawn in Section 5.
Section snippets
Preliminaries
In this section, we shall first recall some definitions and well known results which will be employed throughout the paper. For more details, readers can refer to [2], [12], [27]. Throughout this paper, () denotes respectively the set of all real (all positive real) numbers.
Definition 2.1 [12] A function is known as a fuzzy number if it satisfies the following conditions: u is normal, i.e., there is an such that ; u is fuzzy convex, i.e., for all and we have
Main results
In this section, firstly, we discuss the existence and uniqueness of positive fuzzy solutions for Eq. (1.1). Secondly, the qualitative characteristics of the corresponding systems for crisp difference equations are presented. Finally, the dynamical behaviors of Eq. (1.1) are investigated independently under two different cases in term of g-division of fuzzy numbers.
Numerical examples
In this section, we give two numerical examples validating our theoretical results.
Example 4.1 Consider the second-order exponential-type fuzzy difference equation where and the initial values are triangular fuzzy numbers, namely From (4.2), we get From (4.3) and (4.4), we get
Conclusion
In this work, by virtue of g-division, we investigate a second-order exponential-type fuzzy difference equation . The existence of positive solution and the qualitative behaviors of Eq. (1.1) are studied. The main results are as follows.
- (1)
Under Case I, the positive fuzzy solution of Eq. (1.1) is bounded and persistent. Furthermore, if (3.24) and (3.25) hold true, then every positive fuzzy solution of Eq. (1.1) tends to a unique equilibrium x as .
- (2)
Under Case II, the positive
Declaration of Competing Interest
The authors declare that they have no competing interests.
Acknowledgements
The authors would like to thank the Editor and the anonymous Reviewers for their helpful comments and valuable suggestions to improve the paper. The work is partially supported by National Natural Science Foundation of China (11761018, 11361012), Priority Projects of Science Foundation at Guizhou University of Finance and Economics (2018XZD02), Platform Talents of Guizhou Provincial Department of Science and Technology ([2017]5736-003), Guizhou Team of Scientific and Technological Innovation
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