An approximation method for continuous pseudocontractive mappings

• Main Authors: Chen Rudong, Song Yisheng
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2006/28950

<p/> <p>Let <inline-formula><graphic file="1029-242X-2006-28950-i1.gif"/></inline-formula> be a closed convex subset of a real Banach space <inline-formula><graphic file="1029-242X-2006-28950-i2.gif"/></inline-formula>, <inline-formula><graphic file="1029-242X-2006-28950-i3.gif"/></inline-formula> is continuous pseudocontractive mapping, and <inline-formula><graphic file="1029-242X-2006-28950-i4.gif"/></inline-formula> is a fixed <inline-formula><graphic file="1029-242X-2006-28950-i5.gif"/></inline-formula>-Lipschitzian strongly pseudocontractive mapping. For any <inline-formula><graphic file="1029-242X-2006-28950-i6.gif"/></inline-formula>, let <inline-formula><graphic file="1029-242X-2006-28950-i7.gif"/></inline-formula> be the unique fixed point of <inline-formula><graphic file="1029-242X-2006-28950-i8.gif"/></inline-formula>. We prove that if <inline-formula><graphic file="1029-242X-2006-28950-i9.gif"/></inline-formula> has a fixed point and <inline-formula><graphic file="1029-242X-2006-28950-i10.gif"/></inline-formula> has uniformly G&#226;teaux differentiable norm, such that every nonempty closed bounded convex subset of <inline-formula><graphic file="1029-242X-2006-28950-i11.gif"/></inline-formula> has the fixed point property for nonexpansive self-mappings, then <inline-formula><graphic file="1029-242X-2006-28950-i12.gif"/></inline-formula> converges to a fixed point of <inline-formula><graphic file="1029-242X-2006-28950-i13.gif"/></inline-formula> as <inline-formula><graphic file="1029-242X-2006-28950-i14.gif"/></inline-formula> approaches to 0. The results presented extend and improve the corresponding results of Morales and Jung (2000) and Hong-Kun Xu (2004).</p>