Abstract
Let be a real Hilbert space, a nonempty closed convex subset of , and a maximal monotone operator with . Let be the metric projection of onto . Suppose that, for any given , , and , there exists satisfying the following set-valued mapping equation: for all , where with as and is regarded as an error sequence such that . Let be a real sequence such that as and . For any fixed , define a sequence iteratively as for all . Then converges strongly to a point as , where .
Highlights
Introduction and preliminariesLet H be a real Hilbert space with the inner product ·, · and norm ·
A classical method to solve the following set-valued equation: 0 ∈ T z, 1.5 where T : Ω → 2H is a maximal monotone operator, is the proximal point algorithm which, starting with any point x0 ∈ H, updates xn 1 iteratively conforming to the following recursion: xn ∈ xn 1 βnT xn 1, ∀n ≥ 0, 1.6 where {βn} ⊂ β, ∞, β > 0, is a sequence of real numbers
Is it possible to establish some strong convergence theorems under the weaker assumption on the error sequence {en} given in 1.7 ?. It is our purpose in this paper to give an affirmative answer to Question 2 under a weaker assumption on the error sequence {en} in Hilbert spaces
Summary
Let H be a real Hilbert space with the inner product ·, · and norm ·. 0 ∈ T z, 1.5 where T : Ω → 2H is a maximal monotone operator, is the proximal point algorithm which, starting with any point x0 ∈ H, updates xn 1 iteratively conforming to the following recursion: xn ∈ xn 1 βnT xn 1, ∀n ≥ 0, 1.6 where {βn} ⊂ β, ∞ , β > 0, is a sequence of real numbers. Guler 3 constructed an example showing that Rockafellar’s proximal point algorithm 1.7 does not converge strongly, in general This gives rise to the following question. It is our purpose in this paper to give an affirmative answer to Question 2 under a weaker assumption on the error sequence {en} in Hilbert spaces For this purpose, we collect some lemmas that will be used in the proof of the main results .
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
