Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation. (English) Zbl 1049.39008
Linear difference equations with fuzzy coefficients were introduced by E. Deeba, A. De Korvin and E. Koh [J. Difference Equ. Appl. 2, No. 4, 365–374 (1996; Zbl 0882.39002)]. This paper examines a difference equation
\[
x_{n+1} = \sum_{i=0}^{k} \frac{A_{i}}{x_{n-i}^{p_i}}
\]
with fuzzy coefficients \(A_i : (0,\infty) \to [0,1]\), which are fuzzy numbers [cf. R. Goetschel and W. Voxman, Fuzzy Sets Syst. 18, 31–43 (1986; Zbl 0626.26014)], where \(p_i >0\), \(i=0,1,\dots,k\). It is a generalization of recent authors’ paper [G. Papaschinopoulos and B. K. Papadopoulos, Soft Comput. 6, No. 6, 456–461 (2002; Zbl 1033.39014)]. Fuzzy solutions are described by solution sequences of the following system of recurrence equations
\[
y_{n+1} = \sum_{i=0}^{k} \frac{b_{i}}{z_{n-i}^{p_i}}, \quad z_{n+1} = \sum_{i=0}^{k} \frac{c_{i}}{y_{n-i}^{p_i}}
\]
with positive coefficients. This leads to a description of asymptotic behaviour of fuzzy solutions based on the stability theory of nonlinear recurrence equations [cf. V. L. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications (1993; Zbl 0787.39001)].
Reviewer: Józef Drewniak (Rzeszów)
MSC:
| 39A11 | Stability of difference equations (MSC2000) |
| 26E50 | Fuzzy real analysis |
| 39A20 | Multiplicative and other generalized difference equations |
Keywords:
fuzzy number; \(\alpha\)-cuts; fuzzy solutions; system of recurrence equations; fuzzy difference equations; boundedness; persistence; asymptotic behavior; stabilityReferences:
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