On the convergence and representation of random fuzzy number integrals
Let \(X\) be a separable reflexive Banach space. A function \(u: X\to[0,1]\) is said to be a Lipschitz fuzzy number if \(u^\alpha:= \{x\in X: u(x)\geq \alpha\}\) is a nonempty closed bounded convex subset of \(X\) for \(\alpha\in ]0,1]\) and there is a constant \(L> 0\) such that \(H(u^\alpha, u^\beta)\leq L|\alpha- \beta|\) for all \(\alpha, \beta\in [0,1]\), where \(H\) denotes the Hausdorff metric. The authors introduce in the space \({\mathcal F}_L(X)\) of all Lipschitz fuzzy numbers on \(X\) the concept of w-s-convergence: \(u_n@>\text{w-s}>> u\) means that \(H(x^* (u^\alpha_n), x^*(u^\alpha))\to 0\) \((n\to\infty)\) for any \(x^*\in X^*\) and \(\alpha\in [0,1]\). w-s-convergence is investigated for sequences in \({\mathcal F}_L(X)\) and for sequences of integrals with values in \({\mathcal F}_L(X)\). Moreover, it is studied the relationship between integration and differentiation for functions \(F: [a,b]\to {\mathcal F}_L(X)\).
scientific article; zbMATH DE number 1339548