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The Hundal example revisited. (English) Zbl 1065.47048

H. Hundal [Nonlinear Anal., Theory Methods Appl. 57, No. 1, 35–61 (2004; Zbl 1070.46013)] solved an open problem by constructing an ingenious example of cone and a half-space in the Hilbert space l 2 which intersect at the origin, such that the corresponding sequence of alternating nearest point mappings does not convergence in norm to zero.

In the present paper, the authors, using geometric insight, modify Hundal’s example and simplify its verification. In modified example, the cone and the half-space may be disjoint. They also establish several weak and strong convergence theorems for products and convex combinations of retractions. These results provide the background for the Hundal example and its modifications.

MSC:
47H09Mappings defined by “shrinking” properties
41A65Abstract approximation theory
47H30Particular nonlinear operators
47J25Iterative procedures (nonlinear operator equations)
47N10Applications of operator theory in optimization, convex analysis, programming, economics
90C25Convex programming