The Bruck's ergodic iteration method for the Ky Fan inequality over the fixed point set

ABSTRACT

We introduce a new iteration algorithm for solving the Ky Fan inequality over the fixed point set of a nonexpansive mapping, where the cost bifunction is monotone without Lipschitz-type continuity. The algorithm is based on the idea of the ergodic iteration method for solving multi-valued variational inequality which is proposed by Bruck [On the weak convergence of an ergodic iteration for the solution of variational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), pp. 159–164] and the auxiliary problem principle for equilibrium problems P.N. Anh, T.N. Hai, and P.M. Tuan. [On ergodic algorithms for equilibrium problems, J. Glob. Optim. 64 (2016), pp. 179–195]. By choosing suitable regularization parameters, we also present the convergence analysis in detail for the algorithm and give some illustrative examples.

KEYWORDS:

• Ky fan inequalities
• monotonicity
• fixed points
• nonexpansive mappings

AMS 2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 65 K10
• 90 C25

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by Ministry of Education of the republic of Korea [2015R1D1A1A09058177] and the second author is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under [grant number 101.02-2017.03]. This work was completed during the stay of the second author at the Department of Mathematics Education, Kyungnam University, Korea, 2016.