Approximation of relaxed solutions for lower semicontinuous differential inclusions. (English) Zbl 0755.34015

Ornelas, A.

Ann. Pol. Math. 56, No. 1, 1-10 (1991).

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

Keywords:

differential inclusions; relaxed solutions; continuous selections