On the structure of the solution set of evolution inclusions with time-dependent subdifferentials
The Cauchy problem \[ \begin{gathered} -x'(t)\in \partial \varphi\bigl(t, x(t)\bigr) +F \bigl(t,x(t) \bigr),\;t\in T=[0,b], \\ x(0)=\xi, \tag{P} \end{gathered} \] is considered in a real separable Hilbert space \(H\). Here \(\xi\in H\), \(\varphi: T\times H \to(-\infty, \infty]\) is proper, convex and lower semicontinuous on \(H\) for each fixed \(t\), and satisfies a time-dependence condition of \textit{S. Yotsutani} type [J. Math. Soc. Japan 31, 623-646 (1978; Zbl 0414.35041)], while the multifunction \(F:T\times H\to 2^H\) is assumed to be closed valued. The authors first discuss the nonemptiness and the topological structure of the solution set of (P) when \(\varphi(t,.)\) is of compact type for every \(t\in T\), and \(F\) has convex values. The study is then extended to the case when \(F\) is no longer convex-valued and the functions \(\varphi(t,.)\) are not necessarily of compact type. Some examples involving parabolic distributed parameter systems are also presented.
scientific article; zbMATH DE number 1089776
scientific article