Behavior of the positive solutions of fuzzy max-difference equations

Abstract

We extend some results obtained in 1998 and 1999 by studying the periodicity of the solutions of the fuzzy difference equations xn+1=max{A/xn,A/xn−1,…,A/xn−k}, xn+1=max{A0/xn,A1/xn−1}, where k is a positive integer, A, Ai, i=0,1, are positive fuzzy numbers, and the initial values xi, i=−k,−k+1,…,0 (resp., i=−1,0) of the first (resp., second) equation are positive fuzzy numbers

The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation

“…, À1. Arguing as in Proposition 3.1 of [26] we can easily prove that (L n,a , R n,a ), n = 0,1,. . .…”

“…Also, this fuzzy equation is a generalizing form of the corresponding fuzzy equation of (1) (see [26]). More precisely, we consider the fuzzy difference equation of the form…”

“…The second is that the initial value may be imprecise and uncertain and so the model may contain uncertain parameters. Since fuzzy logic can handle various types of vagueness, but particularly vagueness related to human linguistic and thinking, a fuzzy approach is required (for the theory of fuzzy logic see [12,18] and for fuzzy difference equations see [5,6,14,15,[19][20][21][23][24][25][26]30]). On the other hand, max operators are often used in the study of automatic control's problems (see [9,10,17,22] and the references cited therein) and therefore, max-difference equations have been attracting increasing attention in recent years (see [2,11,16,27,28]).…”

“…where A, B are positive real constants, x À2 , x À1 , x 0 , are positive real numbers, and later Voulov (see [28]) studied the generalizing form Finally, in [26] we considered and studied the corresponding fuzzy difference equations of Eqs. (1) and (2).…”

Dynamics of the fuzzy difference equation z n = max { 1 z n − m , α n z n − r }

“…In 2005, Stefanidou and Papaschinopoulos [13] studied the periodicity of the following fuzzy maxdifference equations z n+1 = max{ α z n , α z n−1 , . .…”