On the strong convergence of sequences of Halpern type in Hilbert spaces. (2nd November 2018)

• Record Type: Journal Article
• Authors: Jaipranop, Ch.; Saejung, S.
• Is Part Of: Optimization. Volume 67:Number 11(2018)
• Page Start: 1895
• Page End: 1922
• Publication Date: 2018-11-02
• Subjects: Fixed point -- sequence of Halpern type -- averaging matrix -- concentrating matrix -- L-hybrid mapping; 47H09 -- 47J20 -- 47J25; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2018.1512108
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: In this paper, we introduce a concept of A -sequences of Halpern type where A is an averaging infinite matrix. If A is the identity matrix, this notion become the well-know sequence generated by Halpern's iteration. A necessary and sufficient condition for the strong convergence of A -sequences of Halpern type is given whenever the matrix A satisfies some certain concentrating conditions. This class of matrices includes two interesting classes of matrices considered by Combettes and Pennanen [J. Math. Anal. Appl. 2002;275:521–536]. We deduce all the convergence theorems studied by Cianciaruso et al. [Optimization. 2016;65:1259–1275] and Muglia et al. [J. Nonlinear Convex Anal. 2016;17:2071–2082] from our result. Moreover, these results are established under the weaker assumptions. We also show that the same conclusion remains true under a new condition.