An existence principle for a periodic solution of a differential equation in Banach space

The equation $d^2x/dt^2=Ax+f(t,x)$ is considered in a Banach space $E$, where $A$ is a fixed unbounded linear operator, and $f(t,x)$ is a nonlinear operator which is periodic in $t$ and satisfies a Lipschitz condition with respect to $x\in E$. Existence conditions have been obtained for a well defined generalized periodic solution of this equation, and also when this solution coincides with the true solution. Similar results have been obtained for the first order equation.