In this paper, the following periodic system is studied: \[ a(u'(t))' \in A(u(t)) + f(t,u(t),u'(t)) \quad \text{a.e.}\;\, t \in [0,b],\qquad u(0)=u(b), \ u'(0)=u'(b), \] where \(a : \mathbb{R}^N \to \mathbb{R}^N\) is continuous and strictly monotone, \(A\) is a maximal monotone operator with \(D(A) \subset \mathbb{R}^N\), and the map \(f\) is a Carathéodory function. Among the assumptions, an unilateral contraint is imposed on \(f\). The existence of a solution is established. The result relies on the Leray-Schauder alternative and on the theory of maximal monotone operators.