Approximating the solutions of differential inclusions driven by measures
The paper focuses on measure differential inclusions of the form \[ \mathrm{d}x(t)\in F(t,x(t))\mathrm{d}\mu(t),\quad t\in[0,1], \] where \(F\) is defined on \([0,1]\times\mathbb R^d\), its values are compact convex subsets of \(\mathbb R^d\), and \(\mu\) is a Stieltjes measure generated by a left-continuous nondecreasing function. A solution of the inclusion is a function \(x:[0,1]\to\mathbb R^d\) for which there exists a \(g:[0,1]\to\mathbb R^d\) such that \(g(t)\in F(t,x(t))\) \(\mu\)-almost everywhere in \([0,1]\), and \[ x(t)=x(0)+\int_0^t g(s)\,\mathrm{d}\mu(s),\quad t\in[0,1]. \] The main results are a global existence theorem yielding the existence of a bounded variation solution, and two theorems dealing with continuous dependence of solutions with respect to the choice of the measure \(\mu\). The results are based on the concept of a bounded \(\varepsilon\)-variation introduced in [\textit{D. Fraňková}, Math. Bohem. 116, No. 1, 20--59 (1991; Zbl 0724.26009)]. In particular, it is assumed that the right-hand side \(F\) satisfies a condition that corresponds to bounded \(\varepsilon\)-variation for multifunctions.
scientific article; zbMATH DE number 7137055
scientific article