An Extension of Gregus Fixed Point Theorem

• Main Authors: Olaleru JO, Akewe H
• Format: Article
• Language: English
• Published: SpringerOpen 2007-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2007/078628

<p/> <p>Let <inline-formula><graphic file="1687-1812-2007-078628-i1.gif"/></inline-formula> be a closed convex subset of a complete metrizable topological vector space <inline-formula><graphic file="1687-1812-2007-078628-i2.gif"/></inline-formula> and <inline-formula><graphic file="1687-1812-2007-078628-i3.gif"/></inline-formula> a mapping that satisfies <inline-formula><graphic file="1687-1812-2007-078628-i4.gif"/></inline-formula> for all <inline-formula><graphic file="1687-1812-2007-078628-i5.gif"/></inline-formula>, where <inline-formula><graphic file="1687-1812-2007-078628-i6.gif"/></inline-formula>, <inline-formula><graphic file="1687-1812-2007-078628-i7.gif"/></inline-formula>, <inline-formula><graphic file="1687-1812-2007-078628-i8.gif"/></inline-formula>, <inline-formula><graphic file="1687-1812-2007-078628-i9.gif"/></inline-formula>, <inline-formula><graphic file="1687-1812-2007-078628-i10.gif"/></inline-formula>, and <inline-formula><graphic file="1687-1812-2007-078628-i11.gif"/></inline-formula>. Then <inline-formula><graphic file="1687-1812-2007-078628-i12.gif"/></inline-formula> has a unique fixed point. The above theorem, which is a generalization and an extension of the results of several authors, is proved in this paper. In addition, we use the Mann iteration to approximate the fixed point of <inline-formula><graphic file="1687-1812-2007-078628-i13.gif"/></inline-formula>.</p>