The author proves a sufficient condition for the global existence of bounded \(C^0\)-solutions for a class of nonlinear functional differential evolution equations of the form \[ u'(t)\in Au(t)+f(t), \,\, t\in [0,+\infty ), \] \[ f(t)\in F(t,u(t),u(t-\tau_1),\dots,u(t-\tau_n)), \,\, t\in [0,+\infty ) \] \[ u(t)=g(u)(t), \,\, t\in [-\tau ,0], \] where \(X\) is a real Banach space, \(A\) is the infinitesimal generator of a nonlinear compact semigroup, \(F:\mathbb R_+\times [\overline{D(A)}]^{n+1}\rightsquigarrow X\) is a nonempty, convex, weak compact valued, and almost strongly-weakly u.s.c. multi-function, and \(g:C_b([-\tau, +\infty);\overline{D(A)}\to C([-\tau ,0];\overline{D(A)})\) is nonexpansive.