Periodicity and stability of solutions to neutral differential equations with phi-Lipschitz and infinite delays

摘要

We prove the existence and uniqueness of a periodic solution to neutral function partial differential equations with infinite delay of the form partial derivative Fu(t)/partial derivative t = A(t)Fu(t) + g(t, u(t)) for t > 0, u(t) = phi(t) for t = 0) of linear partial differential operators is such that the mapping t bat right arrow A(t) is 1-periodic, and the nonlinear delay operator g (t , u) is 1-periodic with respect to t and phi-Lipschitz with respect to u (i.e. parallel to g(t , u) - g(t, v)parallel to = 0) generates an evolution family which has an exponential dichotomy. Moreover, we prove the existence of a local stable manifold around such a periodic solution.