The authors deal with the existence of solutions to an evolution inclusion of the form \[ x'(t) +A(t,x(t))\in{F(t,x(t))}\quad\text{a.e., }x(0)=x_{0}, \] in a Banach space, where the right-hand side is not necessarily convex-valued. It is an improvement of results by \textit{N. S. Papageorgiou} [Dyn. Syst. Appl. 2, No. 1, 61-74 (1993; Zbl 0780.34045)]. Actually, the paper is divided into four sections. After presenting the introduction and some necessary preliminaries the authors investigate the existence of extreme solutions. Next, they show that these solutions form a dense subset in the solution set with convexified right-hand side. In the fifth section the authors derive a version of the bang-bang principle for control systems and finally, the paper closes with an example of a nonlinear parabolic control problem.