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On a non-convex hyperbolic differential inclusion. (English) Zbl 0769.34018

The author shows the existence of a global solution for the Darboux problem \(u_{xy}\in F(x,y,u)\), \(u(x,0)=\alpha(x)\), \(u(0,y)=\beta(y)\), where \(\alpha\), \(\beta\) are two continuous functions from \([0,1]\) into \(\mathbb{R}^ n\) with \(\alpha(0)=\beta(0)\), \(F\) is a multifunction defined on \([0,1]\times [0,1] \times\mathbb{R}^ n\) with decomposable values, measurable with respect to \((x,y)\) and Lipschitzian with respect to \(u\). The key to proving this result is a selection theorem for multifunctions with decomposable values due to Fryszkowski and Bressan-Colombo.

MSC:

34A60 Ordinary differential inclusions
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[11] DOI: 10.1137/0305040 · Zbl 0238.34010
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