Asymptotic almost periodicity to some evolution equations
After introducing a new property of semi-Lipschitz continuity for a class of asymptotic almost periodic functions and without imposing the Lipschitz condition on the nonlinear term, the authors study the existence of asymptotically almost periodic mild solution to some class of autonomous semilinear evolution equations of the form: \[ u'(t) = A u(t) + f(t,u(t)),\quad t \in \mathbb{R} \] in a Banach space \(\mathbb{X},\) where \(A:D(A)\subset \mathbb{X} \to \mathbb{X}\) is the infinitesimal generator of a hyperbolic \(C_{0}-\)semigroup, \(f: \mathbb{R} \times \mathbb{X} \to \mathbb{X}\) is semi-Lipschitz continuous, and \[ \frac{d}{dt}\big[u(t) - h(t,u(t))\big] = Au(t) + f(t, u(t)),\quad t \in \mathbb{R}, \] where \(A:D(A)\subset \mathbb{X} \to \mathbb{X}\) is the infinitesimal generator of a compact analytic semigroup such that \(0\in \rho(A)\) and \(h: \mathbb{R} \times \mathbb{X} \to \mathbb{X_{\alpha}}\) satisfies some appropriate conditions.
scientific article; zbMATH DE number 6510074