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The asymptotic behavior of the composition of two resolvents. (English) Zbl 1075.47033

Let $A$ and $B$ be two maximal monotone operators from a Hilbert space $ℋ$ to ${2}^{ℋ}$ with resolvents ${J}_{A}$ and ${J}_{B},$ respectively, and let $\gamma \in \right]0,\infty \left[·$ The paper under review is concerned with the inclusion problem

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

and its dual

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

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.

##### 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