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

Staicu, Vasile

Proc. Edinb. Math. Soc., II. Ser. 35, No. 3, 375-382 (1992).

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.

Reviewer: J.Myjak (L’Aquila)

MSC:

34A60	Ordinary differential inclusions
35B30	Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Keywords:

differential inclusions; decomposable mapping; global solution; Darboux problem; multifunction; selection theorem

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