Existence of fixed points of fuzzy mappings via theory of fuzzy differential equations
Let \(I= [t_0,t_0+ a]\), \(t_0\geq 0\), \(a>0\) and denote by \(E^n\) the set of all mappings \(u: \mathbb{R}^n\to[0, 1]\) such that \(u\) is normal, \(u\) is fuzzy convex, \(u\) is upper semi-continuous and \([u]^0= \{x\in\mathbb{R}^n: u(x)> 0\}\) is compact. Consider the autonomous initial value problems \[ u'= f(u),\quad u(0)= u_0,\quad\text{with }f\in C(E^n, E^n).\tag{\(*\)} \] The aim of this paper is the following theorem: Assume that \(\lim_{h\to 0+}\sup(d(u+ hf(u), v+ hf(v))- d(u,v))\leq -\beta d(u,v)\) for some \(\beta>0\), with \([u]^\alpha= \{x\in\mathbb{R}^n: u(x)\geq \alpha\}\), \(0<\alpha\leq 1\), \(d_H(A,B)\) is the Hausdorff distance between the sets \(A\), \(B\) of \(P_k(\mathbb{R}^n)= \{H: H\subset \mathbb{R}^n\), \(H\neq \phi\), \(H\) compact, \(H\) convex\}, \(d(u,v)= \sup\{d_H([u]^\alpha,[v]^\alpha): 0\leq\alpha\leq 1\}\); \(D(f(u),\widehat O)\leq M\) if \(d(u,\widehat O)\leq L\), where \(\widehat O\in E^n\) as \(\widehat O(x)= 0\) if \(x= 0\), \(\widehat O(x)\neq 0\) if \(x\neq 0\); for each \(x_0\in E^n\) there exists a solution locally on \([0,a]\) to problems \((*)\). Then, for each \(x_0\in E^n\) there is a unique solution on \([0,\infty)\) to problems \((*)\).
scientific article; zbMATH DE number 1398404