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.