The author constructs \(\tau\)-anti-periodic solutions \((u(t+\tau)=-u(t))\) to the systems \(u'+\partial G(u)=f\) and \(u''+B(u')+\partial G(u)=f,\) where f is anti-periodic, \(\partial G\) is the gradient of an even function \(G: {\mathbb{R}}^ N\to {\mathbb{R}}^ N,\) \(B: {\mathbb{R}}^ N\to {\mathbb{R}}^ N\) is monotone, odd and \((Bz,z)\geq \alpha | z|^ 2-C\) for some constants \(\alpha >0\), \(C\geq 0\). These results are generalized to the abstract equation \(u'+\partial \phi (u)-\lambda u=f,\) where \(\partial \phi\) is the subdifferential of an even, proper convex lower semi-continuous functional \(\phi\) on a real Hilbert space H, with applications to parabolic problems. Existence results are obtained also for the nonlinear wave equation \[ u_{tt}-\Delta u+f(u)+\beta (u_ t)\ni h\quad in\quad R\times \Omega,\quad u=0\quad on\quad R\times \partial \Omega,\quad \Omega \subset R^ N, \] where \(\beta\) is an odd maximal monotone operator and the function f is odd and satisfies additional growth conditions.