Iterative approaches to convex feasibility problems in Banach spaces

Abstract The convex feasibility problem (CFP) of finding a point in the nonempty intersection ⋂ i = 1 N C i is considered, where N ⩾ 1 is an integer and each C i is assumed to be the fixed point set of a nonexpansive mapping T i : X → X 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 is strongly convergent to a solution of (CFP) provided the Banach space X either is uniformly smooth or is reflexive and has a weakly continuous duality map, and provided 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.

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