On the fuzzy difference equation xn = F(xn−1,xn−k)☆
Introduction
Recently there has been an increase in interest in the study of linear and nonlinear fuzzy difference equations (see, e.g. [3], [5], [8], [10], [15], [17], [20], [21], [22], [24]) and max-type fuzzy difference equation (see, e.g. [4], [14], [16], [18]) because many models in biology, ecology, physiology physics, engineering, economics, probability theory, genetics, psychology and resource management are represented by these equations naturally (see, e.g. [1], [2], [6], [9]). For example, Chrysafis et al. [1] demonstrated the applicability of the fuzzy difference equations in fields of science such as finance. They also study the time value of money via fuzzy difference equations and provide simplicity in the analysis of more complicated cases which concern the balance of a bank deposit as a special application of the time value of money. In [2], Deeba and Korvin studied the second-order linear difference equation where and the initial values are fuzzy numbers (here denotes the set for any integers ). This fuzzy equation is a linearized model of a nonlinear model which determines the carbon dioxide level in the blood.
In [11], Stefanidou and Papaschinopoulos studied the existence, the boundedness and the asymptotic behavior of the positive solutions of the following fuzzy difference equation where , A and the initial values are positive fuzzy numbers.
In 2015, Zhang et al. [19] investigated the boundedness, persistence and global behavior of the positive solutions of the following third-order rational fuzzy difference equation where A and the initial values are positive fuzzy numbers.
In [13], Rahmana et al. studied the qualitative behavior of following second-order fuzzy rational difference equation where and the initial values are positive fuzzy numbers.
In 2018, Wang et al. [23] investigated the existence and uniqueness of the positive solutions and the asymptotic behavior of the equilibrium points of the fuzzy difference equation where and the initial values are positive fuzzy numbers.
Motivated by the above studies, in this paper, we consider the more general fuzzy difference equation where and the initial values are positive fuzzy numbers and F is the Zadeh's extension (or the fuzzification) of f and f satisfies the following hypotheses:
, where .
is strictly increasing on x and strictly decreasing on y.
There exists a strictly decreasing function , where , such that
- (i)
For any , and ;
- (ii)
and .
- (i)
The rest of this paper is organized as follows. In Section 2 we give some definitions and notations. In Section 3 we give the main result and its proof of this paper. In Section 4 we present some examples to illustrate the applicability of the main result of this paper.
Section snippets
Preliminaries
For the convenience of the readers, we give the following definitions and notations.
Definition 2.1 Let X be a normed space. Denote by the set of all functions u from X to interval which satisfies the following conditions (i)-(iv): u is normal (i.e., there exists an such that ). u is fuzzy convex (i.e., for all and ). u is upper semi-continuous. The support of u, is compact, where denote the closure of subset B of
Main result
In the sequel, let be a positive solution of (1.1) with initial values . For any , write Then we see from Proposition 2.1 that satisfies the following system with initial values .
From Proposition 2.2 we see that there exists an with such that for any and any , Lemma 3.1 Let
Examples
To illustrate the applicability of Theorem 3.1, we present the following examples.
Example 4.1 Consider fuzzy difference equation where , the initial values and are trivial fuzzy numbers. Let It is easy to verify that hold for f. By Theorem 3.1 we see that every positive solution of (4.1) converges to a positive equilibrium of (4.1) as .
Example 4.2 Consider fuzzy difference equation
Conclusion
In this paper, we study the more general fuzzy difference equation (1.1) and obtain sufficient conditions under which every positive solution of (1.1) converges to a positive equilibrium. Furthermore, we present three examples to illustrate the applicability of the main result of this paper. For further research, we plan to study the more general fuzzy difference equation .
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Project Supported by NNSF of China (11761011) and NSF of Guangxi (2018GXNSFAA294010, 2016GXNSFAA380286) and SF of Guangxi University of Finance and Economics (2018QNA03).