Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

We consider the nonlinear delay differential evolution equation $$left{begin{array}{ll} u'(t) in Au(t) + f(t, u_t), quad quad t in mathbb{R}_+, u(t) = g(u)(t),qquad qquad quad t in [-tau, 0], end{array} right.$$ u ′ ( t ) ∈ A u ( t ) + f ( t , u t ) , t ∈ R + , u ( t ) = g ( u ) ( t ) , t ∈ [ - τ , 0 ] , where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as ${t mapsto {rm{e}}^{-omega t}}$ t ? e - ω t when ${t to + infty}$ t → + ∞ and ${f : {mathbb{R}}_+ times C([-tau, 0]; overline{D(A)}) to X}$ f : R + × C ( [ - τ , 0 ] ; D ( A ) ¯ ) → X is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant ? satisfies the condition ${ell{rm{e}}^{omegatau} < omega, g : C_b([-tau, +infty); overline{D(A)}) to C([-tau, 0]; overline{D(A)})}$ ? e ω τ < ω , g : C b ( [ - τ , + ∞ ) ; D ( A ) ¯ ) → C ( [ - τ , 0 ] ; D ( A ) ¯ ) is nonexpansive and (I – A)^?1 is compact, then the unique C ⁰-solution of the problem above is almost periodic.