This paper is concerned with stability properties of a Lur'e system obtained by interconnection of a general linear time-invariant block (possibly, infinite-dimensional) and a periodic nonlinearity. Such systems usually have multiple equilibria. In the paper, two new frequency-algebraic stability criteria are established by using. Popov's method of "a priori integral indices", Leonov's method of nonlocal reduction and the Bakaev-Guzh technique.