Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces

Volume 9, Issue 6, pp 3702--3718 http://dx.doi.org/10.22436/jnsa.009.06.21

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Authors

Yaqiang Liu - School of Management, Tianjin Polytechnic University, Tianjin 300387, China.. Shin Min Kang - Department of Mathematics and the RINS, Gyeongsang National University, Jinju 52828, Korea.. Youli Yu - School of Mathematics and Information Engineering, Taizhou University, Linhai 317000, China.. Lijun Zhu - School of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan 750021, China..

Abstract

In this paper, we introduce two general algorithms (one implicit and one explicit) for finding a common element of the set of an equilibrium problem and the set of common fixed points of a nonexpansive semigroup \(\{T(s)\}_{s\geq 0}\) in Hilbert spaces. We prove that both approaches converge strongly to a common element \(x^*\) of the set of the equilibrium points and the set of common fixed points of \(\{T(s)\}_{s\geq 0}\). Such common element \(x^*\) is the unique solution of some variational inequality, which is the optimality condition for some minimization problem. As special cases of the above two algorithms, we obtain two schemes which both converge strongly to the minimum norm element of the set of the equilibrium points and the set of common fixed points of \(\{T(s)\}_{s\geq 0}\). The results obtained in the present paper improve and extend the corresponding results by Cianciaruso et al. [F. Cianciaruso, G. Marino, L. Muglia, J. Optim. Theory. Appl., 146 (2010), 491-509] and many others.

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Keywords

• Equilibrium problem
• variational inequality
• fixed point
• nonexpansive semigroup
• algorithm
• minimum norm.

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