Convergence of Paths for Perturbed Maximal Monotone Mappings in Hilbert Spaces

• Main Authors: Zhou Haiyun, Kang ShinMin, Qing Yuan, Qin Xiaolong
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Fixed Point Theory and Applications
• Online Access: http://www.fixedpointtheoryandapplications.com/content/2010/547828
• ISSN: 1687-1820; 1687-1812

<p/> <p>Let <inline-formula> <graphic file="1687-1812-2010-547828-i1.gif"/></inline-formula> be a Hilbert space and <inline-formula> <graphic file="1687-1812-2010-547828-i2.gif"/></inline-formula> a nonempty closed convex subset of <inline-formula> <graphic file="1687-1812-2010-547828-i3.gif"/></inline-formula>. Let <inline-formula> <graphic file="1687-1812-2010-547828-i4.gif"/></inline-formula> be a maximal monotone mapping and <inline-formula> <graphic file="1687-1812-2010-547828-i5.gif"/></inline-formula> a bounded demicontinuous strong pseudocontraction. Let <inline-formula> <graphic file="1687-1812-2010-547828-i6.gif"/></inline-formula> be the unique solution to the equation <inline-formula> <graphic file="1687-1812-2010-547828-i7.gif"/></inline-formula>. Then<inline-formula> <graphic file="1687-1812-2010-547828-i8.gif"/></inline-formula> is bounded if and only if <inline-formula> <graphic file="1687-1812-2010-547828-i9.gif"/></inline-formula> converges strongly to a zero point of <it>A</it> as <inline-formula> <graphic file="1687-1812-2010-547828-i10.gif"/></inline-formula> which is the unique solution in <inline-formula> <graphic file="1687-1812-2010-547828-i11.gif"/></inline-formula>, where <inline-formula> <graphic file="1687-1812-2010-547828-i12.gif"/></inline-formula> denotes the zero set of <inline-formula> <graphic file="1687-1812-2010-547828-i13.gif"/></inline-formula>, to the following variational inequality <inline-formula> <graphic file="1687-1812-2010-547828-i14.gif"/></inline-formula>, for all <inline-formula> <graphic file="1687-1812-2010-547828-i15.gif"/></inline-formula>.</p>