Stability of solution sets of differential equations with multivalued right-hand side. (English) Zbl 0808.34013

Constantin, Adrian

J. Differ. Equations 114, No. 1, 243-252 (1994).

The following initial value problem in \(\mathbb{R}^ n\) for a differential inclusion is considered: \(x'(t) \in F(t,x(t))\), \(x(0) = b\). It is assumed that \(F\) is continuous with convex and compact values and \(D(F(t,x)\), \(F(t,y)) \leq k(t) | x - y |\) with a continuous function \(k (D(A,B)\) stands for the Hausdorff distance of the sets \(A,B)\). Then it is proved that the map \(S\) which associates to every initial condition \(b \in \mathbb{R}^ n\) the solution set of the above differential inclusion is continuous from \(\mathbb{R}^ n\) to the space of all continuous maps \(x : \mathbb{R}_ + \to \mathbb{R}^ n\) such that \(x(t)/g(t)\) is bounded on \(\mathbb{R}_ +\), where \(g(t) = \exp (2 \int^ t_ 0 k(s)ds)\).

Reviewer: V.Křivan (České Budějovice)

MSC:

34A60	Ordinary differential inclusions
34D20	Stability of solutions to ordinary differential equations

Keywords:

continuous; dependence on initial condition; initial value problem; differential inclusion