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:xn+1=Axn−1xn−2/B+∑i=3kCixn−i n=0,1,2,…,wherexnis the sequence of positive fuzzy numbers and the parametersA,B,C3,C4,…,Ckand initial conditionsx0,x−1,x−2,x−ii=3,4,…,kare positive fuzzy numbers. Besides, some numerical examples describing the fuzzy difference equation are given to illustrate the theoretical results.
Highlights
It is well known that difference equations are one of the most widely used equations in various subject areas
Where xn is a sequence of the positive fuzzy numbers and the parameters A, B, C3, C4, . . . , Ck and initial conditions x0, x−1, x−2, x−i(i 3, 4, . . . , k) are the positive fuzzy numbers. e purpose of this paper is to study the asymptotic behavior of the equilibrium point of the fuzzy difference equation. e main method is to convert the fuzzy difference equation into a rational difference equation according to the fuzzy number theory, and the properties of the solutions of the fuzzy difference equations are obtained by studying the corresponding constant difference equations
A variational iterative method for the nonlinear fuzzy difference equations is proposed. is method is a powerful tool for solving various fuzzy difference equations and can be applied to other nonlinear differential equations or difference equations in mathematical physics
Summary
It is well known that difference equations are one of the most widely used equations in various subject areas. 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: xn−1 , xn−1xn−2 n 0, 1, 2, . In 2017, the author in [12] investigated the asymptotic behavior of the equilibrium points for the following fuzzy difference equation: D. where xn is a sequence of the positive fuzzy numbers and the parameters A, B, C, D and initial conditions x−4, x−3, x−2, x−1, x0 are the positive fuzzy numbers. E 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. 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) e sufficient conditions obtained are new, general, and verifiable, which provide flexibility for the application and analysis of the high-order fuzzy difference equation
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