A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems
- 28 Downloads
We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz-type condition as well as Hölder continuity of the bifunctions involved.
KeywordsMonotone equilibria Fixed point Common solution Splitting algorithm
Mathematics Subject Classification47H05 47H10 90C33
The authors would like to thank the associate editor and anonymous referee for their constructive comments and helpful remarks. This work is supported by the National Foundation for Science and Technology Development (NAFOSTED) of Vietnam under grant number 101.01-2017.315.
- 11.Fan, K.: A minimax inequality and applications. In: Shisha, O. (ed.) Inequality, vol. III, pp. 103–113. Academic Press, New York (1972)Google Scholar
- 13.Hieu, D.V., Moudafi, A.: A barycentric projected-subgradient algorithm for equilibrium problems. J. Nonlinear Var. Anal. 1(1), 43–59 (2017)Google Scholar
- 32.Sun, S.: An alternative regularization method for equilibrium problems and fixed point of nonexpansive mappings. J. Appl. Math. 2012, Article ID 202860 (2012). https://doi.org/10.1155/2012/202860