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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 to 2 with resolvents J A and J B , respectively, and let γ]0,[· The paper under review is concerned with the inclusion problem

find(x,y) 2 suchthat(0,0)(Id-R+γ(A×B))(x,y),(1)

and its dual

find(x * ,y * ) 2 suchthat(0,0)((Id-R) -1 +(A -1 ×B -1 )(Id/γ))(x * ,y * )·(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.

MSC:
47J05Equations involving nonlinear operators (general)
47H09Mappings defined by “shrinking” properties
90C25Convex programming
49J40Variational methods including variational inequalities
47H05Monotone operators (with respect to duality) and generalizations
47J25Iterative procedures (nonlinear operator equations)
47N10Applications of operator theory in optimization, convex analysis, programming, economics