On evolution equations having monotonicities of opposite sign (Q2640012)

The main results of the present paper deal with the existence of local solutions to the Cauchy problem \((1)\quad x'(t_ 0)\in -\partial V(x(t))+F(x(t)),\quad x(0)=x_ 0,\) where \(\partial V\) denotes a subdifferential of a real-valued function V and F is a multifunction with compact values. Using an iterative method the author has proved the following existence theorem: Theorem. Let V: \({\mathbb{R}}^ n\to (-\infty,+\infty]\) be a proper convex lower semicontinuous function and \(x_ 0\in {\mathbb{R}}^ n\) be such that \(\partial V(x_ 0)\neq \emptyset\); let F be an upper semicontinuous cyclically monotone map with compact nonempty values defined on a neighborhood of \(x_ 0\). Then there exists \(T>0\) and x: [0,T]\(\to {\mathbb{R}}^ n\), a solution to the Cauchy problem (1).