Periodic Solution for a Max-Type Fuzzy Difference Equation

Changyou Wang,Jiahui Li

doi:10.1155/2020/3094391

Changyou Wang, Jiahui Li

Open Access

https://doi.org/10.1155/2020/3094391

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Abstract

The paper is concerned with the dynamics behavior of positive solutions for the following max-type fuzzy difference equation system: xn+1=maxA/xn, A/xn−1, xn−2, n=0,1,2,…, where xn is a sequence of positive fuzzy numbers, and the parameter A and the initial conditions x−2, x−1, x0 are also positive fuzzy numbers. Firstly, the fuzzy set theory is used to transform the fuzzy difference equation into the corresponding ordinary difference equations with parameters. Then, the expression for the periodic solution of the max-type ordinary difference equations is obtained by the iteration, the inequality technique, and the mathematical induction. Moreover, we can obtain the expression for the periodic solution of the max-type fuzzy difference equation. In addition, the boundedness and persistence of solutions for the fuzzy difference equation is proved. Finally, the results of this paper are simulated and verified by using MATLAB 2016 software package.

Highlights

With the continuous development of science and technology in the fields of economy, biology, computer science, and so on, the research of nonlinear difference equations has been rapidly pushed forward
Making a historical flash back for the max-type difference equation we study in this paper, we should mention that, in 2002, Voulov [11] studied positive solutions of the following equation: AB
Where k is a positive integer, A is a real constant, and 􏼈xi􏼉0i − k are real numbers. He proved that every positive solution of (3) is eventually periodic of period k + 2

Summary

Introduction

With the continuous development of science and technology in the fields of economy, biology, computer science, and so on, the research of nonlinear difference equations has been rapidly pushed forward, (for example, see [1,2,3,4] and the relevant reference cited therein). Where k is a positive integer, A is a real constant, and 􏼈xi􏼉0i − k are real numbers He proved that every positive solution of (3) is eventually periodic of period k + 2. In 2004, Stefanidou and Papaschinopoulos [42] extended equation (3) from real number to fuzzy number, where A is a positive fuzzy number and the initial conditions 􏼈xi􏼉0i − k are positive fuzzy numbers, and they gave a condition so that the solution is eventually periodic, unbounded, and nonpersistent and considered and studied the corresponding fuzzy difference equation (9) in [42] when k 0, m 1, i.e., xn+1. In 2006, Stefanidou and Papaschinopoulos [43], to further extend the difference equation (7), considered the periodic nature of the positive solutions of the following fuzzy difference equation: max􏼨 A0 , xn− k. Where A are positive fuzzy numbers and the initial values x− 2, x− 1, x0 are any positive fuzzy numbers. is paper aims to study the periodicity of the positive solutions of (9) by using a new iteration method for the more general nonlinear difference equations and inequality skills, as well as the mathematical induction

Preliminaries and Notations

Main Results

An Illustrative Example