Second order nonlinear multivalued boundary problems in Hilbert spaces (Q1771013)

The authors consider the following second-order multivalued boundary value problem \[ \begin{cases} p(t) x''(t)+r(t) x'(t)\in Ax(t)+F(t,x(t)), &\text{a.e}\,\, t\in [0,T],\\ (x'(0),-\tilde r(T)x'(T))\in \xi(x(0)-a,x(T)-b), \end{cases} \] where \(F:[0,T]\times H\to 2^{H}\setminus\emptyset\) is a multifunction, \(A\) is a maximal monotone operator in a Hilbert space \(H,\) \(\xi\) is maximal monotone in \(H\times H,\) \(a, b\in D(A),\) \(p,r\in C([0,T],\mathbb R)\) and \(\tilde r(t)=\exp\left(\int_0^tr(s)/p(s)\, ds\right).\) Using techniques from multivalued analysis and the theory of nonlinear operators of monotone type, they prove the existence of solutions for both the ``convex'' and ``nonconvex'' ~problems. An example illustrating the results is also presented.