KKM–Fan’s lemma for solving nonlinear vector evolution equations with nonmonotonic perturbations

In this paper we are interested in the existence of solutions of the following initial value problem: du/dt+Au+Gu=f on (0,T)(0,T) with u(0)=u0u(0)=u0 where A:V→V′A:V→V′ is a monotone operator, G:V→V′G:V→V′ is a nonlinear nonmonotone operator and f:(0,T)→V′f:(0,T)→V′ is a measurable function, by means of a recent generalization of the famous KKM–Fan’s lemma.