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Vietnam Journal of Mathematics 40:2&3(2012)
371-389
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Strong Convergence of Two Hybrid Extragradient
Methods
for Solving Equilibrium and Fixed Point Problems
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Jean Jacques Strodiot1,2, Van
Hien Nguyen1,2 and Phan Tu Vuong1
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1Institute for Computational Science and
Technology at Ho Chi Minh City
(ICST HCMC),
Vietnam
2University of Namur (FUNDP), Belgium
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Dedicated to Professor
Phan Quoc Khanh on the occasion of his 65th birthday
Received December 18,
2011
Revised February 7,
2012
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Abstract. In this paper we propose and we study two
algorithmic methods for finding a common solution of an equilibrium problem
and a fixed point problem in a Hilbert space. The strategy is to replace
the proximal point iteration used in most papers by an extragradient
procedure with or without an Armijo-backtracking linesearch. The strong
convergence of the iterates generated by each method is obtained thanks to
a shrinking projection method and under the assumptions that the fixed
point mapping is a $\xi$-quasi-strict pseudo-contraction and the
equilibrium function is monotone and Lipschitz-continuous for the pure
extragradient method and pseudomonotone and weakly continuous for the
extragradient method with linesearches. The particular case when the equilibrium
problem is a variational inequality problem is considered in the last
section.
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2000 Mathematics
Subject Classification. 47H06, 47H09, 47J05, 47J25.
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Keywords.
Equilibrium problem, fixed point problem, shrinking projection method,
extragradient method, $\xi$-quasi-strict pseudo-contraction, Lipschitz
continuity.
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Established
by Vietnam Academy of Science and Technology &
Vietnam Mathematical Society
Published by
Springer since January 2013
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