Approximate Proximal Point Algorithms for Finding Zeroes of Maximal Monotone Operators in Hilbert Spaces

• Main Authors: Kang ShinMin, Zhou Haiyun, Cho YeolJe
• Format: Article
• Language: English
• Published: SpringerOpen 2008-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2008/598191
• ISSN: 1025-5834; 1029-242X

<p>Abstract</p> <p>Let <inline-formula> <graphic file="1029-242X-2008-598191-i1.gif"/></inline-formula> be a real Hilbert space, <inline-formula> <graphic file="1029-242X-2008-598191-i2.gif"/></inline-formula> a nonempty closed convex subset of <inline-formula> <graphic file="1029-242X-2008-598191-i3.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i4.gif"/></inline-formula> a maximal monotone operator with <inline-formula> <graphic file="1029-242X-2008-598191-i5.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i6.gif"/></inline-formula> be the metric projection of <inline-formula> <graphic file="1029-242X-2008-598191-i7.gif"/></inline-formula> onto <inline-formula> <graphic file="1029-242X-2008-598191-i8.gif"/></inline-formula>. Suppose that, for any given <inline-formula> <graphic file="1029-242X-2008-598191-i9.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2008-598191-i10.gif"/></inline-formula>, and <inline-formula> <graphic file="1029-242X-2008-598191-i11.gif"/></inline-formula>, there exists <inline-formula> <graphic file="1029-242X-2008-598191-i12.gif"/></inline-formula> satisfying the following set-valued mapping equation: <inline-formula> <graphic file="1029-242X-2008-598191-i13.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i14.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i15.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2008-598191-i16.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i17.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i18.gif"/></inline-formula> is regarded as an error sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i19.gif"/></inline-formula>. Let <inline-formula> <graphic file="1029-242X-2008-598191-i20.gif"/></inline-formula> be a real sequence such that <inline-formula> <graphic file="1029-242X-2008-598191-i21.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i22.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2008-598191-i23.gif"/></inline-formula>. For any fixed <inline-formula> <graphic file="1029-242X-2008-598191-i24.gif"/></inline-formula>, define a sequence <inline-formula> <graphic file="1029-242X-2008-598191-i25.gif"/></inline-formula> iteratively as <inline-formula> <graphic file="1029-242X-2008-598191-i26.gif"/></inline-formula> for all <inline-formula> <graphic file="1029-242X-2008-598191-i27.gif"/></inline-formula>. Then <inline-formula> <graphic file="1029-242X-2008-598191-i28.gif"/></inline-formula> converges strongly to a point <inline-formula> <graphic file="1029-242X-2008-598191-i29.gif"/></inline-formula> as <inline-formula> <graphic file="1029-242X-2008-598191-i30.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2008-598191-i31.gif"/></inline-formula><inline-formula> <graphic file="1029-242X-2008-598191-i32.gif"/></inline-formula>.</p>