Bounded mild solutions for semilinear integro differential equations in Banach spaces (Q601719)

The authors study the structure of several classes of spaces of vector-valued functions \(\mathcal{M}(\mathbb{R};X)\); here \(X\) denotes a Banach space. The integro-differential equation \[ u'(t)=Au(t)+\int_{-\infty}^t a(t-s)Au(s)ds+f(t,u(t)) \tag{1} \] is considered, where \(A\) is a closed linear operator defined in \(X\) and \(a\in L^1_{\text{loc}}(\mathbb{R}_+)\) is a scalar-valued kernel. Using a unified approach for various spaces \(\mathcal{M}(\mathbb{R};X)\), the authors establish conditions on \(A\) and \(f\) ensuring that the solution \(u\) of (1) exists and has the same asymptotic behaviour as \(f\). In particular, almost automorphic, pseudo-almost automorphic, asymptotically periodic and almost periodic classes of functions are investigated. Moreover, asymptotically compact almost automorphic functions and pseudo compact almost automorphic functions are introduced in the paper.