Browder's type strong convergence theorems for infinite families of nonexpansive mappings in Banach spaces

• Main Author: Suzuki Tomonari
• Format: Article
• Language: English
• Published: SpringerOpen 2006-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2006/59692

<p/> <p>We prove Browder's type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let <inline-formula><graphic file="1687-1812-2006-59692-i1.gif"/></inline-formula> be a bounded closed convex subset of a uniformly smooth Banach space <inline-formula><graphic file="1687-1812-2006-59692-i2.gif"/></inline-formula>. Let <inline-formula><graphic file="1687-1812-2006-59692-i3.gif"/></inline-formula> be an infinite family of commuting nonexpansive mappings on <inline-formula><graphic file="1687-1812-2006-59692-i4.gif"/></inline-formula>. Let <inline-formula><graphic file="1687-1812-2006-59692-i5.gif"/></inline-formula> and <inline-formula><graphic file="1687-1812-2006-59692-i6.gif"/></inline-formula> be sequences in <inline-formula><graphic file="1687-1812-2006-59692-i7.gif"/></inline-formula> satisfying <inline-formula><graphic file="1687-1812-2006-59692-i8.gif"/></inline-formula> for <inline-formula><graphic file="1687-1812-2006-59692-i9.gif"/></inline-formula>. Fix <inline-formula><graphic file="1687-1812-2006-59692-i10.gif"/></inline-formula> and define a sequence <inline-formula><graphic file="1687-1812-2006-59692-i11.gif"/></inline-formula> in <inline-formula><graphic file="1687-1812-2006-59692-i12.gif"/></inline-formula> by <inline-formula><graphic file="1687-1812-2006-59692-i13.gif"/></inline-formula> for <inline-formula><graphic file="1687-1812-2006-59692-i14.gif"/></inline-formula>. Then <inline-formula><graphic file="1687-1812-2006-59692-i15.gif"/></inline-formula> converges strongly to <inline-formula><graphic file="1687-1812-2006-59692-i16.gif"/></inline-formula>, where <inline-formula><graphic file="1687-1812-2006-59692-i17.gif"/></inline-formula> is the unique sunny nonexpansive retraction from <inline-formula><graphic file="1687-1812-2006-59692-i18.gif"/></inline-formula> onto <inline-formula><graphic file="1687-1812-2006-59692-i19.gif"/></inline-formula>.</p>