Abstract
In this paper, we propose two novel parallel hybrid methods for finding a common element of the set of solutions of a finite family of generalized equilibrium problems for monotone bifunctions \(\left\{ f_i\right\} _{i=1}^N\) and \(\alpha \)-inverse strongly monotone operators \(\left\{ A_i\right\} _{i=1}^N\) and the set of common fixed points of a finite family of (asymptotically) \(\kappa \)-strictly pseudocontractive mappings \(\left\{ S_j\right\} _{j=1}^M\) in Hilbert spaces. The strong convergence theorems are established under the standard assumptions imposed on equilibrium bifunctions and operators. Some numerical examples are presented to illustrate the efficiency of the proposed parallel methods.
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Notes
The matrices \(P_i,~Q_i\) are randomly generated as follows: we randomly choose \(\lambda _{1k}^i\in [-m,0],~\lambda _{2k}^i\in [1,m],~ k=1,\ldots ,m,~i=1\ldots ,N\). Set \(\widehat{Q}_1^i\), \(\widehat{Q}_2^i\) as two diagonal matrices with eigenvalues \(\left\{ \lambda _{1k}^i\right\} _{k=1}^m\) and \(\left\{ \lambda _{2k}^i\right\} _{k=1}^m\), respectively. Then, we make positive definite matrices \(Q_i\) and negative semidefinite matrices \(T_i\) by using random orthogonal matrices with \(\widehat{Q}_2^i\) and \(\widehat{Q}_1^i\), respectively. Finally, set \(P_i=Q_i-T_i\), (\(i=1,\ldots ,N\)).
References
Alber, Y.I.: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartosator, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Applied Mathematics, vol. 178, pp. 15–50. Dekker, New York (1996)
Anh, P.K., Buong, Ng, Hieu, D.V.: Parallel methods for regularizing systems of equations involving accretive operators. Appl. Anal. 93(10), 2136–2157 (2014)
Anh, P.K., Chung, C.V.: Parallel hybrid methods for a finite family of relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 35(6), 649–664 (2014)
Anh, P.K., Hieu, D.V.: Parallel and sequential hybrid methods for a finite family of asymptotically quasi \(\phi \)-nonexpansive mappings. J. Appl. Math. Comput. 48, 241–263 (2015)
Anh, P.K., Hieu, D.V.: Parallel hybrid methods for variational inequalities, equilibrium problems and common fixed point problems. Vietnam J. Math. (2015). doi:10.1007/s10013-015-0129-z
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20, 197–228 (1967)
Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6(1), 117–136 (2005)
Cioranescu, I.: Geometry of Banach spaces, duality mappings and nonlinear problems. In: Rosen, K.H., Krithivasan, K. (eds.) Mathematics and Its Applications, vol. 62. Kluwer Academic Publishers, Dordrecht (1990)
Duan, P.: Convergence theorems concerning hybrid methods for strict pseudocontractions and systems of equilibrium problems. J. Inequal. Appl., Vol 2010 (2010). Article ID 396080. doi:10.1155/2010/396080
Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171–174 (1972)
Hieu, D.V.: A parallel hybrid method for equilibrium problems, variational inequalities and nonexpansive mappings in Hilbert space. J. Korean Math. Soc. 52, 373–388 (2015)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody 12, 747–756 (1976)
Kim, J.K., Cho, S.Y., Qin, X.: Some results on generalized equilibrium problems involving strictly pseudocontractive mappings. Acta Math. Sci. 31B(5), 2041–2057 (2011)
Liu, Q., Zeng, W., Huang, N.: An iterative method for generalized equilibrium problems, fixed point problems and variational inequality problems. Fixed Point Theory Appl. Vol 2009 (2009). Article ID 531308. doi:10.1155/2009/531308
Mann, W.R.: Mean value methods in iteration. Proc. Am. Math. Soc. 4, 506–510 (1953)
Muu, L.D., Oettli, W.: Convergence of an adative penalty scheme for finding constrained equilibria. Nonlinear Anal. 18(12), 1159–1166 (1992)
Nakajo, K., Takahashi, : Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J. Math. Anal. Appl 279, 372–379 (2003)
Nirenberg, L.: Topics in nonlinear functional analysis [Russian translation], Mir, Moscow (1977)
Qihou, L.: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 26, 1835–1842 (1996)
Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Reinermann, J.: Uber fixpunkte kontrahievuder Abbidungen und schwach konvergente Tooplite-Verfahren. Arch. Math. 20, 59–64 (1969)
Rhoades, B.E.: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 56(3), 741–750 (1976)
Sahu, D.R., Xu, H.K., Yao, J.C.: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal. 70, 3502–3511 (2009)
Zhang, H., Su, Y.: Strongly convergence theorem fro generalized equilibrium problems and countable family of nonexpansive mappings. Int. J. Optim. Theory Methods Appl. 1(1), 87–101 (2009)
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Van Hieu, D. Parallel hybrid methods for generalized equilibrium problems and asymptotically strictly pseudocontractive mappings. J. Appl. Math. Comput. 53, 531–554 (2017). https://doi.org/10.1007/s12190-015-0980-9
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DOI: https://doi.org/10.1007/s12190-015-0980-9