The asymptotic behavior of the composition of two resolvents (Q707223)

Let \(A\) and \(B\) be two maximal monotone operators from a Hilbert space \({\mathcal H}\) to \(2^{{\mathcal H}}\) with resolvents \(J_{A}\) and \(J_{B},\) respectively, and let \(\gamma\in ]0,\infty[.\) The paper under review is concerned with the inclusion problem \[ \text{find } (x,y)\in {\mathcal H}^{2} \text{ such that }(0,0)\in (\text{Id}-R+\gamma(A\times B))(x,y),\tag{1} \] and its dual \[ \text{find }(x^{*},y^{*})\in {\mathcal H}^{2} \text{ such that } (0,0)\in ((\text{Id}-R)^{-1}+ (A^{-1}\times B^{-1})\circ (\text{Id}/\gamma))(x^{*},y^{*}).\tag{2} \] Connections are made between the solutions of (1) and (2). The applications provided include variational inequalities, the problem of finding cycles for inconsistent feasibility problems, a study of an alternating minimization procedure and a new proof of von Neumann's classical result on the method of alternating projections.