Existence of periodic solutions for nonlinear differential equations (Q1273691)

The author is concerned with the existence of solutions to the following second-order abstract differential equation with T-periodic boundary condition: \[ x_{tt}+ Ax_t- A^*x_t- A^*Ax+ B(t)x+ f(t,x(t))= 0\quad\text{on }\mathbb{R}, \] \[ x(0)= x(T),\quad x_t(0)= x_t(T). \] One reduces this problem to a system and then one uses duality and variational methods in complex Hilbert spaces.