Existence of solutions to a class of evolution inclusions
The authors study the existence of local solutions to the following evolution system \[ x'\in -\partial _{F}V(x)+F(x)+f(t,x),\quad x(0)=x_{0},\tag{1} \] where \(\partial _{F}V\) is the Fréchet subdifferential of a function \(V\) with a \(\psi -\)monotone subdifferential of order \(2\), \(F\) is an upper semicontinuous, cyclically monotone multifunction, and \(f:[0,+\infty)\times \Omega \rightarrow \mathbb{R}^{n}\) is a Carathéodory function, where \(\Omega \) is an open subset of \(\mathbb{R}^{n}.\) Their result extends previous results by \textit{A. Cellina} and \textit{V. Staicu} [J. Differ. Equations 90, No. 1, 71-80 (1991; Zbl 0719.34030)], \textit{V. Staicu} [Rend. Semin. Mat., Torino 48, No. 2, 137-148 (1990; Zbl 0756.34022)], and \textit{F. Papalini} [J. Math. Anal. Appl. 193, No. 2, 406-418 (1995; Zbl 0835.34022)] who considered, respectively, the following special cases of (1): \(x'\in -\partial V(x)+F(x),\) \(x(0)=x_{0}\); \((2)\) \(x'\in -\partial V(x)+F(x)+f(t,x)\), \(x(0)=x_{0}\); \((3)\) \(x'\in -\partial _{F}V(x)+F(x),\) \(x(0)=x_{0}\); and \((4)\) where \(\partial V\) is the subdifferential of a proper convex lower semicontinuous function \(V,\) and \(F\) and \(f\) are as in (1).
scientific article; zbMATH DE number 1788054
scientific article