The asymptotic behavior of the composition of two resolvents.

*(English)*Zbl 1075.47033Let $A$ and $B$ be two maximal monotone operators from a Hilbert space $\mathscr{H}$ to ${2}^{\mathscr{H}}$ with resolvents ${J}_{A}$ and ${J}_{B},$ respectively, and let $\gamma \in ]0,\infty [\xb7$ The paper under review is concerned with the inclusion problem

$$\text{find}\phantom{\rule{4.pt}{0ex}}(x,y)\in {\mathscr{H}}^{2}\phantom{\rule{4.pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}(0,0)\in (\text{Id}-R+\gamma (A\times B))(x,y),\phantom{\rule{2.em}{0ex}}\left(1\right)$$

and its dual

$$\text{find}\phantom{\rule{4.pt}{0ex}}({x}^{*},{y}^{*})\in {\mathscr{H}}^{2}\phantom{\rule{4.pt}{0ex}}\text{such}\phantom{\rule{4.pt}{0ex}}\text{that}\phantom{\rule{4.pt}{0ex}}(0,0)\in ({(\text{Id}-R)}^{-1}+({A}^{-1}\times {B}^{-1})\circ (\text{Id}/\gamma ))({x}^{*},{y}^{*})\xb7\phantom{\rule{2.em}{0ex}}\left(2\right)$$

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.

Reviewer: Sotiris K. Ntouyas (Ioannina)

##### MSC:

47J05 | Equations involving nonlinear operators (general) |

47H09 | Mappings defined by “shrinking” properties |

90C25 | Convex programming |

49J40 | Variational methods including variational inequalities |

47H05 | Monotone operators (with respect to duality) and generalizations |

47J25 | Iterative procedures (nonlinear operator equations) |

47N10 | Applications of operator theory in optimization, convex analysis, programming, economics |