Anti-periodic solutions for evolution equations associated with monotone type mappings

In this work we study the anti-periodic problem { x ′ ( t ) ∈ − ∂ ϕ x ( t ) − ∂ G x ( t ) + f ( t ) , a.e.  t ∈ R , x ( t ) = − x ( t + T ) , t ∈ R in a separable Hilbert space where ϕ : D ( ϕ ) ⊆ H → ( − ∞ , + ∞ ] is an even lower semi-continuous convex function, G : H → R is an even continuous differentiable function such that ∂ G is a demi-continuous mapping of class ( S + ) or pseudo-monotone and f : R → H is a continuous mapping satisfying f ( t + T ) = − f ( t ) for t ∈ R . Two existence results are obtained.