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Iterative approaches to convex feasibility problems in Banach spaces. (English) Zbl 1139.47056

Summary: The convex feasibility problem (CFP) of finding a point in the nonempty intersection i=1 N C i is considered, where N1 is an integer and each C i is assumed to be the fixed point set of a nonexpansive mapping T i :XX with X a Banach space. It is shown that the iterative scheme

x n+1 =λ n+1 y+(1-λ n+1 )T n+1 x n ,

where T k =T kmodN if k>N, is strongly convergent to a solution of (CFP) provided that the Banach space X either is uniformly smooth or is reflexive and has a weakly continuous duality map, and provided that the sequence {λ n } satisfies certain conditions. The limit of {x n } is located as Q(y), where Q is the sunny nonexpansive retraction from X onto the common fixed point set of the T i ’s.


MSC:
47N10Applications of operator theory in optimization, convex analysis, programming, economics
90C25Convex programming
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
90C48Programming in abstract spaces