Strong and weak solutions to second order differential inclusions governed by monotone operators (Q742200)

Let \(A\) be a maximal monotone operator in a Hilbert space \(H;\) a function \(f: (0, \mathbb R) \to H\) belongs to the class \(X_R\), i.e., \(\int_0^Rt\|f(t)\|dt < \infty.\) A function \(u \in C([0,T];H)\) is called a weak solution to the problem \[ u^{\prime \prime} (t) \in Au(t) + f(t) \quad \text{a.e. \,\,on} \, (0,\mathbb R), \tag{1} \] \[ u(0) = x, \,u(R) = y \tag{2} \] if there exist sequences \(u_n \in W^{2,2}(0,\mathbb R;H)\) and \(f_n \in L^2(0,\mathbb R; H)\) such that (i) \(u_n\) satisfies (1) with \(f = f_n;\) (ii) \(f_n\) converges to \(f\) in \(X_R;\) (iii) \(u_n\) converges to \(u\) in \(C([0,\mathbb R];H);\) (iv) \(u\) satisfies (2). The authors prove the existence and uniqueness of a weak solution \(u\) to (1), (2) such that \(t^{1/2}u^{\prime} \in L^2(0,\mathbb R;H).\) The existence and uniqueness of a bounded weak solution \(u\) to inclusion (1) on the half axis \(\mathbb{R}_+\) is also considered. It is mentioned that if \(f \in L_{\mathrm{loc}}^2(0,\infty;H)\) then \(u\) is a strong solution.