# zbMATH — the first resource for mathematics

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
Full Text: