On the strong convergence of sequences of Halpern type in Hilbert spaces

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.

KEYWORDS:

• Fixed point
• sequence of Halpern type
• averaging matrix
• concentrating matrix
• L-hybrid mapping

AMS CLASSIFICATIONS:

• 47H09
• 47J20
• 47J25

Acknowledgments

The authors would like to thank the referee for comments and suggestions on the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

S. Saejung  http://orcid.org/0000-0003-3325-2864

Additional information

Funding

The first author is supported by the Science Achievement Scholarship of Thailand (SAST). The second author is supported by the Thailand Research Fund and Khon Kaen University under grant RSA5980006.