Directionally continuous selections and nonconvex evolution inclusions
In [\textit{A. Bressan} and \textit{V. Staicu}, Set-Valued Anal. 2, No. 3, 415-437 (1994; Zbl 0820.47072)], the evolution inclusion in a Banach space \[ \dot x\in Ax+ F(t,x)\tag{1} \] is considered, in which \(A\) is assumed to generate a contractive semigroup on \(\text{cl}(D(A))\), \(F\) is assumed to be bounded, lower semicontinuous and closed-valued (but not necessarily convex-valued) and the domain of \(F\), denoted by \(\Omega\subseteq\mathbb{R}\times \text{cl}(D(A))\), is assumed to be compact. There is introduced a special topology \(\tau\) (stronger than the metric one) on \(\mathbb{R}\times \text{cl}(D(A))\) and then, a theorem due to Bressan and Cortes is applied to obtain a selection \(f\) of \(F\) which is directionally continuous with respect to the topology \(\tau\). Then, a set-valued function \(G\), which is upper semicontinuous and compact- and convex-valued, is constructed from \(f\). Finally, it is shown that the solutions to the evolution inclusion \[ \dot x\in Ax+ G(t,x)\tag{2} \] are solutions to (1). In this way, existence of solutions and qualitative properties of solutions to (1) can be deduced from those for (2). In the article under review, the authors show that the topology \(\tau\) satisfies a certain property which allows them to apply a different theorem to obtain the solution \(f\), namely the one due to \textit{A. Bressan} and \textit{G. Colombo} [Stud. Math. 102, No. 3, 987-992 (1992; Zbl 0807.54020)]. In this way, the assumption of compactness of \(\Omega\) can be avoided. The authors actually deal here with a Scorza-Dragoni lower semicontinuous \(F\), a weaker assumption than that of lower semicontinuity.
scientific article; zbMATH DE number 1901528