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 ${\ell{\rm{e}}^{\omega\tau} < \omega, g : C_b(-\tau, +\infty); \overline{D(A)}) \to C(-\tau, 0]; \overline{D(A)})}$ ? 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.