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Zbl 1139.47056
O'Hara, John G.; Pillay, Paranjothi; Xu, Hong-Kun
Iterative approaches to convex feasibility problems in Banach spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl. 64, No. 9, A, 2022-2042 (2006). ISSN 0362-546X

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 2000:: *47N10 Appl. of operator theory in optimization, math. programming, etc.
90C25 Convex programming
47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
90C48 Programming in abstract spaces

Keywords: iterative method; strong convergence; convex feasibility problem; nonexpansive mapping; duality mapping; sunny nonexpansive retraction; Banach space

Cited in: Zbl 1140.47057

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