Second-order differential equations on half-line associated with monotone operators (Q1269553)

The boundary value problem \[ p(t)u''(t)+ r(t) u'(t)\in Au(t)+ f(t),\quad\text{a.e. on }[0,\infty),\quad u'(0)\in \alpha(u(0)- a),\tag{1} \] is considered, where \(A\) and \(\alpha\) are maximal monotone operators in a real Hilbert space \(H\), \(a\in D(A)\), and \(f\), \(p\), \(r\) satisfy suitable assumptions. The existence of a solution to (1) is proved, and the continuous dependence on data of solutions is established. The equivalence of the problem with an optimization problem is proved. An interesting example illustrates the abstract theory.