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

Let [inline-graphic not available: see fulltext] be a real Hilbert space, [inline-graphic not available: see fulltext] a nonempty closed convex subset of [inline-graphic not available: see fulltext], and [inline-graphic not available: see fulltext] a maximal monotone operator with [inline-graphic not available: see fulltext]. Let [inline-graphic not available: see fulltext] be the metric projection of [inline-graphic not available: see fulltext] onto [inline-graphic not available: see fulltext]. Suppose that, for any given [inline-graphic not available: see fulltext], [inline-graphic not available: see fulltext], and [inline-graphic not available: see fulltext], there exists [inline-graphic not available: see fulltext] satisfying the following set-valued mapping equation: [inline-graphic not available: see fulltext] for all [inline-graphic not available: see fulltext], where [inline-graphic not available: see fulltext] with [inline-graphic not available: see fulltext] as [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext] is regarded as an error sequence such that [inline-graphic not available: see fulltext]. Let [inline-graphic not available: see fulltext] be a real sequence such that [inline-graphic not available: see fulltext] as [inline-graphic not available: see fulltext] and [inline-graphic not available: see fulltext]. For any fixed [inline-graphic not available: see fulltext], define a sequence [inline-graphic not available: see fulltext] iteratively as [inline-graphic not available: see fulltext] for all [inline-graphic not available: see fulltext]. Then [inline-graphic not available: see fulltext] converges strongly to a point [inline-graphic not available: see fulltext] as [inline-graphic not available: see fulltext], where [inline-graphic not available: see fulltext][inline-graphic not available: see fulltext].