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Fixed points for multifunctions on generalized metric spaces with applications to a multivalued Cauchy problem. (English) Zbl 0938.34005
An existence theorem for the following differential inclusion \[ x'(t) \in F(x(t)), \;x(t_0) = x_0, \] is proved. The basic assumptions are such that \(F\) is defined on the subset \(\Omega \) of \(\mathbb{R} \times \mathbb{R}^n\), its values are compact subsets of \(\mathbb{R}^n\), and it is upper semicontinuous. Moreover, \(|t-t_0|H(F(t,u)\), \(F(t,v)) \leq k\|u-v\|\) and \(|t-t_0|^{\beta} H(F(t,u), F(t,v))\leq A\|u-v\|^{\alpha}\) for every \((t,u), (t,v)\in \Omega \) where \(A,k>0\), \(0<\alpha <1\), \(\beta < \alpha \) and \(k(1-\alpha) < 1-\beta \). \(H\) denotes the Hausdorff distance. The proof is based on a fixed point theorem.
MSC:
34A60 Ordinary differential inclusions
47H10 Fixed-point theorems
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