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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 $\bigcap_{i=1}^{N}C_{i}$ is considered, where $N\geq 1$ is an integer and each $C_{i}$ is assumed to be the fixed point set of a nonexpansive mapping $T_{i}\colon X\rightarrow X$ with $X$ a Banach space. It is shown that the iterative scheme $$x_{n+1}=\lambda_{n+1}\,y+(1-\lambda_{n+1})T_{n+1}\,x_{n},$$ where $T_{k}=T_{k\bmod N}$ 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 $\{\lambda_{n}\}$ 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
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References:
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