Generic properties of multifunctions: application to differential inclusions
Consider the differential inclusion \[ \dot{x}(t)\in F(t,x(t)),\quad x(0) = x_{0},\quad t\in I=[0,1], \tag{1} \] where \(F:I\times \mathbb R^{n} \rightrightarrows \mathbb R^{n}\) has nonempty convex compact values. Let \(\text{ext}\, A:=\{a\in A: a =tb+(1-t) c\), \(b,c\in A\), \(t\in [0,1] \Rightarrow a= b = c\}\) and \[ \dot{x}(t)\in \text{cl}(\text{ext}\, F(t,x(t))),\quad x(0) = x_{0},\quad t\in I. \tag{2} \] The multi-map \(T(\cdot,\cdot)\) is said to be one sided Kamke (OSK) if there exists a Kamke function \(w(\cdot,\cdot)\) such that \[ \sigma(x-y,F(t,x)) -\sigma(x-y,F(t,y))\leq 0.5 w(t,|x-y|^{2}), \] where \(\sigma(l,A)=\sup_{a\in A}\langle l,a\rangle\) is the support function. Theorem. For every differential inclusion in \(\mathbb R^{n}\) with compact convex values and OSK right-hand side satisfying a linear growth conditions, the solution set of (2) is dense in the solution set of (1). The main properties of differential inclusions with Kamke and one sided Kamke right-hand sides are studied in the paper.
scientific article; zbMATH DE number 5865534