Existence and equivalence with an optimization problem for some difference inclusions (Q1882469)

Let \(H\) be a real Hilbert space with a scalar product \((\cdot,\cdot)\). Let \(A:D(A)\subset H\to H\) and \(\alpha:D(A)\subset H\to H\) be maximal monotone operators [see \textit{V. Barbu}, Nonlinear semigroups and differential equations in Banach spaces. Revised and enlarged translation of the Romanian ed. (1976; Zbl 0328.47035) for basic properties] such that \(0\in D(A)\cap D(\alpha)\) and \(0\in\alpha(0)\). Let \(0<\theta_n<1\) for \(n\geq 1\) and \(a_0=1, a_n=(\theta_1\dots\theta_n)^{-1}\) for \(n>1\). The author considers the Hilbert space \({\mathcal L}=\ell^2(H)\) endowed with the scalar product \(\langle (u_n)_n,(v_n)_n\rangle=\sum_{n=1}^\infty a_n(u_n,v_n)\) and studies the following difference inclusion problem \[ u_{n+1}-(1+\theta_n)u_n+\theta_n u_{n-1}\in c_nAu_n+f_n,\quad u_1-u_0\in\alpha(u_0-a), \] where \(a\in H\) is fixed, \((u_n)_n\in{\mathcal L}\), and \(f_n\in H\), \(c_n>0\) for all \(n\geq 1\). He establishes the maximal monotonicity of the operator \(B\) defined by \(B((u_n)_n)=(-u_{n+1}+(1+\theta_n)u_n-\theta_n u_{n-1})_n\) with the domain \(D(B)=\{(u_n)_n\in{\mathcal L},u_1-u_0\in\alpha(u_0-a)\}\). This fact is the key ingredient in the given proof of the existence of solutions of the above difference inclusion problem. Moreover, if \(A\) or \(\alpha\) is one-to-one, then the solution is unique. The author proves also that the above difference inclusion problem is equivalent to some minimization problem. Finally, he considers some examples.