Solutions to partial functional differential equations with infinite delays: periodicity and admissibility
The authors consider partial functional differential equations with infinite delay of the form \[ \dot{u}=A(t)u+g(t,u_{t}),\quad t\in\mathbb{R}^{+} \] with \(u_{t}(s):=u(t+s)\) for \(s\in\left(-\infty,0\right]\) on a Banach space \(X\) having a separable predual \(Y\). First they study the existence and uniqueness of periodic solutions when \(A(t)\) is \(1\)-periodic and the nonlinear term \(g(t,\phi)\) is \(1\)-periodic with respect to \(t\) for every fixed \(\phi\) in memory phase spaces, and is \(\phi(t)\)-Lipschitz. Also, they apply their results to the case when the family \(\left(A(t)\right)_{t\geq0}\) generates an evolution family having an exponential dichotomy, to study the existence, uniqueness and conditional stability of periodic solutions. They conclude with the existence of a local stable manifold near the periodic solution in that case.
partial functional differential equations
scientific article; zbMATH DE number 6909837
scientific article