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The periodic nature of the positive solutions of a nonlinear fuzzy max-difference equation. (English) Zbl 1122.39008

In a series of papers the authors generalized results on particular difference equations from the crisp case to a fuzzy case [cf. G. Papaschinopoulos and B. K. Papadopoulos, Fuzzy Sets Syst. 129, No. 1, 73–81 (2002; Zbl 1016.39015), Soft Comput. 6, No. 6, 456–461 (2002; Zbl 1033.39014), G. Stefanidou and G. Papachinopoulos, Fuzzy Sets Syst. 140, No. 3, 523–539 (2003; Zbl 1049.39008), Adv. Difference Equ. 2004, No. 4, 337–357 (2004; Zbl 1084.39015), J. Nonlinear Math. Phys. 12, Suppl. 2, 300–315, electronic only (2005; Zbl 1088.39008)].
This paper concerns the max-difference equation: \[ x_n = \max(A/x_{n-k},B/x_{n-m}), n \geq 0 \] for fixed \(A, B > 0\), positive integers \(k, m\) and positive initial values. H. D. Voulov [J. Difference. Equ. Appl. 8, No. 9, 799–810 (2002; Zbl 1032.39004)] proved periodicity of positive solution of this equation (eventually periodic) and determined three possible values of periods. Now, a generalization of this equation admits positive fuzzy numbers in the place of coefficients and initial values. However, the generalization of results is not successful: a periodicity is obtained if at least one of coefficients \(A, B\) is a trivial fuzzy number (crisp number). Alternatively, the authors prove that positive fuzzy solutions are unbounded.

MSC:

39A11 Stability of difference equations (MSC2000)
26E50 Fuzzy real analysis
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