Approximating fixed points of the composition of two resolvent operators


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	Approximating fixed points of the composition of two resolvent operators


	Fixed Point Theory, Volume 18, No. 1, 2017, 137-146, March 1st, 2017 DOI: 10.24193/fpt-ro.2017.1.11 Authors: Oganeditse A. Boikanyo Abstract: Let A and B be maximal monotone operators defined on a real Hilbert space H, and let $(J_{\mu}^AJ_{\mu}^B)\neq \emptyset$ , where $J_{\mu}^Ay:=(I \mu A)^{-1}y$ and μ is a given positive number. [H. H. Bauschke, P. L. Combettes and S. Reich, The asymptotic behavior of the composition of two resolvents, Nonlinear Anal. 60 (2005), no. 2, 283-301] proved that any sequence (x_n) generated by the iterative method $x_{n+1}=J_{\mu}^Ay_n$ , with $y_n=J_{\mu}^Bx_n$ converges weakly to some point in $(J_{\mu}^AJ_{\mu}^B)$ . In this paper, we show that the modified method of alternating resolvents introduced in [O. A. Boikanyo, A proximal point method involving two resolvent operators, Abstr. Appl. Anal. 2012, Article ID 892980, (2012)] produces sequences that converge strongly to some points in $(J_{\mu}^AJ_{\mu}^B)$ and $(J_{\mu}^BJ_{\mu}^A)$ . Key Words and Phrases: Maximal monotone operator, alternating resolvents, proximal point algorithm, nonexpansive map, resolvent operator. 2010 Mathematics Subject Classification: 47J25, 47H05, 47H09, 47H10. Published on-line: March 1st, 2017. Abstract pdf Fulltext pdf Back to volume's table of contents