Summary: An evolution inclusion on the half-line in a real Hilbert space is studied, namely \(pu''+ru'\in Au+f\) a.e. on \([0,\infty)\), under the boundary condition \(u'(0)\in\alpha(u(0)-a)\), where \(A\) and \(\alpha\) are maximal monotone operators. Some hypotheses are assumed which permit to establish the existence, uniqueness and continuous dependence on initial data of the solution. The equivalence of this problem to a minimization problem is derived.