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Approximation of relaxed solutions for lower semicontinuous differential inclusions. (English) Zbl 0755.34015
Let \(F:I\times\mathbb{R}^ n\to\mathbb{R}^ n\) be a measurable multifunction, with compact values and integrably bounded, such that \(F(t;\bullet)\) is usc. Define \(G=L^ 1\to L^ 1\) by \(G(x)=\{v\in L^ 1:v(t)\in F(t,x(t))\) a.e.}. It is proved that the set of continuous selections from \(G\) is weakly dense in the set of continuous selections from \(\overline co G\). Using this result, the author then proves that for any selection \(f\in co F\), there is a sequence of solutions \(x_ n\) of \(x\in F(t,x)\) that converges uniformly to some solution \(x_ *\) of \(x=f(t,x)\).
Reviewer: S.Hu (Springfield)

MSC:
34A60 Ordinary differential inclusions
49J45 Methods involving semicontinuity and convergence; relaxation
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