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Sequence spaces of fuzzy real numbers defined by Orlicz functions. (English) Zbl 1199.46167

An Orlicz function \(M\:[0,\infty)\rightarrow [0,\infty)\) is a continuous, convex, non-decreasing function such that \(M(0)=0\) and \(M(x)>0\) for \(x>0\), and \(M(x)\rightarrow \infty \) as \(x\rightarrow \infty \). If the convexity of theOrlicz function \(M\) is replaced by \[ M(x+y)\leq M(x) + M(y), \] then this function is called a modulus function.
The authors define the following sequence spaces of fuzzy real numbers defined by an Orlicz function
\[ (l_\infty)_F(M) = \left \{(X_k): \sup _{k} M \left (\frac {\bar {d}(X_k,\bar {0})}{\rho }\right)< \infty ,\quad \text{for some}\; \rho >0 \right \}, \]
\[ c_F(M) = \left \{(X_k): \lim _{k} M \left (\frac {\bar {d}(X_k,L)}{\rho }\right)=0 ,\quad \text{for some}\; \rho >0\; \text{and}\; L \in R(I)\right \}, \]
\[ (c_0)_F(M) = \left \{(X_k): \lim _{k} M \left (\frac {\bar {d}(X_k,\bar {0})}{\rho }\right)=0 ,\quad \text{for some}\; \rho >0 \right \}. \]

MSC:

46S40 Fuzzy functional analysis
46A45 Sequence spaces (including Köthe sequence spaces)
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