Iterative Approximation to Convex Feasibility Problems in Banach Space

• Main Authors: Yang Li, Kim Jong Kyu, Yao Jen-Chih, Chang Shih-Sen
• Format: Article
• Language: English
• Published: SpringerOpen 2007-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2007/046797

<p/> <p>The convex feasibility problem (CFP) of finding a point in the nonempty intersection <inline-formula><graphic file="1687-1812-2007-046797-i1.gif"/></inline-formula> is considered, where <inline-formula><graphic file="1687-1812-2007-046797-i2.gif"/></inline-formula> is an integer and each <inline-formula><graphic file="1687-1812-2007-046797-i3.gif"/></inline-formula> is assumed to be the fixed point set of a nonexpansive mapping <inline-formula><graphic file="1687-1812-2007-046797-i4.gif"/></inline-formula>, where <inline-formula><graphic file="1687-1812-2007-046797-i5.gif"/></inline-formula> is a reflexive Banach space with a weakly sequentially continuous duality mapping. By using viscosity approximation methods for a finite family of nonexpansive mappings, it is shown that for any given contractive mapping <inline-formula><graphic file="1687-1812-2007-046797-i6.gif"/></inline-formula>, where <inline-formula><graphic file="1687-1812-2007-046797-i7.gif"/></inline-formula> is a nonempty closed convex subset of <inline-formula><graphic file="1687-1812-2007-046797-i8.gif"/></inline-formula> and for any given <inline-formula><graphic file="1687-1812-2007-046797-i9.gif"/></inline-formula> the iterative scheme <inline-formula><graphic file="1687-1812-2007-046797-i10.gif"/></inline-formula> is strongly convergent to a solution of (CFP), if and only if <inline-formula><graphic file="1687-1812-2007-046797-i11.gif"/></inline-formula> and <inline-formula><graphic file="1687-1812-2007-046797-i12.gif"/></inline-formula> satisfy certain conditions, where <inline-formula><graphic file="1687-1812-2007-046797-i13.gif"/></inline-formula> and <inline-formula><graphic file="1687-1812-2007-046797-i14.gif"/></inline-formula> is a sunny nonexpansive retraction of <inline-formula><graphic file="1687-1812-2007-046797-i15.gif"/></inline-formula> onto <inline-formula><graphic file="1687-1812-2007-046797-i16.gif"/></inline-formula>. The results presented in the paper extend and improve some recent results in Xu (2004), O'Hara et al. (2003), Song and Chen (2006), Bauschke (1996), Browder (1967), Halpern (1967), Jung (2005), Lions (1977), Moudafi (2000), Reich (1980), Wittmann (1992), Reich (1994).</p>