Behavior of the positive solutions of fuzzy max-difference equations
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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. 1. Introduction Difference equations are often used in the study of linear and nonlinear physical, physiological, and economical problems (for partial review see [3, 6]). This fact leads to the fast promotion of the theory of difference equations which someone can find, for instance, in [1, 7, 9]. More precisely, max-difference equations have increasing interest since max operators have applications in automatic control (see [2, 11, 17, 18] and the references cited therein). Nowadays, a modern and promising approach for engineering, social, and environmental problems with imprecise, uncertain input-output data arises, the fuzzy approach. This is an expectable effect, since fuzzy logic can handle various types of vagueness but particularly vagueness related to human linguistic and thinking (for partial review see [8, 12]). The increasing interest in applications of these two scientific fields contributed to the appearance of fuzzy difference equations (see [4, 5, 10, 13, 14, 15, 16]). In [17], Szalkai studied the periodicity of the solutions of the ordinary difference equation
A A A , ,... , , xn+1 = max xn xn−1 xn−k
(1.1)
where k is a positive integer, A is a real constant, xi , i = −k, −k + 1,...,0 are real numbers. More precisely, if A is a positive real constant and xi , i = −k, −k + 1,...,0 are positive real numbers, he proved that every positive solution of (1.1) is eventually periodic of period k + 2. Copyright © 2005 Hindawi Publishing Corporation Advances in Difference Equations 2005:2 (2005) 153–172 DOI: 10.1155/ADE.2005.153
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Fuzzy max-difference equations
In [2], Amleh et al. studied the periodicity of the solutions of the ordinary difference equation xn+1 = max
A0 A1 , , xn xn−1
(1.2)
where A0 , A1 are positive real constants and x−1 , x0 are real numbers. More precisely, if A0 , A1 are positive constants, x−1 , x0 are positive real numbers, A0 > A1 (resp., A0 = A1 ) (resp., A0 < A1 ), then every positive solution of (1.2) is eventually periodic of period two (resp., three) (resp., four). In this paper, our goal is to extend the above mentioned results for the corresponding fuzzy difference equations (1.1) and (1.2) where A, A0 , A1 are positive fuzzy numbers and xi , i = −k, −k + 1,...,0, x−1 , x0 are positive fuzzy numbers. Moreover, we find conditions so that the corresponding fuzzy equations (1.1) and (1.2) have unbounded solutions, something that does not happen in case of the ordinary difference equations (1.1) and (1.2). We note that, in order to study the behavior of a parametric fuzzy difference equation we use the following technique: we investigate the behavior of the solutions of a
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