Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

Abstract

We consider the nonlinear delay differential evolution equation $$\left\{\begin{array}{ll} u'(t) \in Au(t) + f(t, u_t), \quad \quad t \in \mathbb{R}_+,\\ u(t) = g(u)(t),\qquad \qquad \quad t \in [-\tau, 0], \end{array} \right.$$ 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 $${t \mapsto {\rm{e}}^{-\omega t}}$$ when $${t \to + \infty}$$ and $${f : {\mathbb{R}}_+ \times C([-\tau, 0]; \overline{D(A)}) \to X}$$ is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant l satisfies the condition $${\ell{\rm{e}}^{\omega\tau} < \omega, g : C_b([-\tau, +\infty); \overline{D(A)}) \to C([-\tau, 0]; \overline{D(A)})}$$ is nonexpansive and (I – A)−1 is compact, then the unique C 0-solution of the problem above is almost periodic.