Anti-periodic solutions for nonlinear evolution inclusions
The authors consider the nonlinear anti-periodic evolution inclusion \[ \begin{aligned} & -u'(t)\in A(t,u(t))+F(t,u(t))\text{ a.e. on }I=[0,T],\\ & u(0)=-u(T),\end{aligned}\eqno{(1)} \] where \(\left( X,H,X^{\ast }\right) \) is an evolution triple, \(A:I\times X\rightarrow 2^{X^{\ast }}\) and \(F:I\times X\rightarrow 2^{H}\). \ It is not assumed that \(A(t,\cdot )\) is maximal monotone or that \(F(t,\cdot )\) is a subdifferential. \ Existence is proven for the case in which \(F\) is convex-valued and also in the case in which it is not necessarily convex-valued. \ Also, existence is proven when \(F\) is replaced by its extreme points. \ It is then shown that the extremal trajectories are dense in the set of solutions of the convex problem. \ An example is given to illustrate this last result. \ The proofs make use of the selection theorem from [\textit{A. Bressan} and \textit{G. Colombo}, Stud. Math. 90, No. 1, 69--86 (1988; Zbl 0677.54013)] and a number of other results from the literature, including the monographs by Hu and Papageorgiou.
scientific article; zbMATH DE number 6932132
scientific article