Splitting methods for a convex feasibility problem in Hilbert spaces

Volume 9, Issue 5, pp 2638--2648 http://dx.doi.org/10.22436/jnsa.009.05.60

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Authors

Yunpeng Zhang - College of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou, 450011, China. Yanling Li - School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power University, Zhengzhou 450011, China.

Abstract

In this paper, we investigate a convex feasibility problem based on a splitting method. Strong convergence theorems are established without the aid of metric projections in the framework of real Hilbert spaces.

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Keywords

• Equilibrium problem
• monotone operator
• feasibility problem
• Hilbert space
• projection.

MSC

•  47H05
•  90C33

References

• [1] J. Balooee, Iterative algorithms for solutions of generalized regularized nonconvex variational inequalities, Non-linear Funct. Anal. Appl., 18 (2013), 127-144.


• [2] V. Barbu, Nonlinear semigroups and differential equations in Banach spaces, Translated from the Romanian, Editura Academiei Republicii Socialiste Romania, Bucharest; Noordhoff International Publishing, Leiden (1976)


• [3] B. A. Bin Dehaish, A. Latif, H. Bakodah, X. Qin, A viscosity splitting algorithm for solving inclusion and equilibrium problems, J. Inequal. Appl., 2015 (2015), 14 pages.


• [4] B. A. Bin Dehaish, X. Qin, A. Latif, O. H. Bakodah, Weak and strong convergence of algorithms for the sum of two accretive operators with applications, J. Nonlinear Convex Anal., 16 (2015), 1321-1336.


• [5] E. Blum, W. Oettli , From optimization and variational inequalities to equilibrium problems, Math. Student, 63 (1994), 123-145.


• [6] F. E. Browder, W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197-228.


• [7] S. Y. Cho, Generalized mixed equilibrium and fixed point problems in a Banach space, J. Nonlinear Sci. Appl., 9 (2016), 1083-1092.


• [8] S. Y. Cho, W. Li, S. M. Kang, Convergence analysis of an iterative algorithm for monotone operators, J. Inequal. Appl., 2013 (2013), 14 pages.


• [9] S. Y. Cho, X. Qin, On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems , Appl. Math. Comput., 235 (2014), 430-438.


• [10] W. Cholamjiak, P. Cholamjiak, S. Suantai, Convergence of iterative schemes for solving fixed point problems for multi-valued nonself mappings and equilibrium problems , J. Nonlinear Sci. Appl., 8 (2015), 1245-1256.


• [11] B. S. Choudhury, S. Kundu, A viscosity type iteration by weak contraction for approximating solutions of generalized equilibrium problem , J. Nonlinear Sci. Appl., 5 (2012), 243-251.


• [12] J. Douglas, H. H. Rachford, On the numerical solution of heat conduction problems in two and three space variables , Trans. Amer. Math. Soc., 82 (1955), 421-439.


• [13] C. S. Hwang, An infinite family nonexpansive mappings and a relaxed cocoercive mapping, Adv. Fixed Point Theory, 4 (2014), 184-198.


• [14] J. K. Kim, P. N. Anh, Y. M. Nam, Strong convergence of an extended extragradient method for equilibrium problems and fixed point problems, J. Korean Math. Soc., 49 (2012), 187-200.


• [15] J. K. Kim, S. Y. Cho, X. Qin, Some results on generalized equilibrium problems involving strictly pseudocontractive mappings, Acta Math. Sci., 31 (2011), 2041-2057.


• [16] L. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces , J. Math. Anal. Appl., 194 (1995), 114-125.


• [17] B. Liu, C. Zhang , Strong convergence theorems for equilibrium problems and quasi-\(\phi\)-nonexpansive mappings, Nonlinear Funct. Anal. Appl., 16 (2011), 365-385.


• [18] S. Lv, Strong convergence of a general iterative algorithm in Hilbert spaces, J. Inequal. Appl., 2013 (2013), 18 pages.


• [19] S. Lv , A new algorithm for solving nonlinear optimization problems, J. Nonlinear Funct. Anal., 2016 (2016), 13 pages.


• [20] X. Qin, S. Y. Cho, J. K. Kim, On the weak convergence of iterative sequences for generalized equilibrium problems and strictly pseudocontractive mappings, Optimization, 61 (2012), 805-821.


• [21] X. Qin, S. Y. Cho, L. Wang, A regularization method for treating zero points of the sum of two monotone operators, Fixed Point Theory Appl., 2014 (2014), 10 pages.


• [22] X. Qin, Y. Su , Approximation of a zero point of accretive operator in Banach spaces, J. Math. Anal. Appl., 329 (2007), 415-424.


• [23] T. Ram , On existence of operator solutions of generalized vector quasi-variational inequalities, Commun. Optim. Theory, 2015 (2015), 8 pages.


• [24] R. T. Rockfellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim., 14 (1976), 877-898.


• [25] T. Suzuki , Strong convergence of Krasnoselskii and Mann's type sequences for one-parameter nonexpansive semi-groups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.


• [26] P. Tseng, A modified forward-backward splitting methods for maximal monotone mappings, SIAM. J Control Optim., 38 (2000), 431-446.


• [27] S. Wang, On fixed point and variational inclusion problems, Filomat, 29 (2015), 1409-1417.


• [28] J. Zhao, Strong convergence theorems for equilibrium problems, fixed point problems of asymptotically nonexpansive mappings and a general system of variational inequalities, Nonliear Funct. Anal. Appl., 16 (2011), 447-464.


• [29] C. Zhang, J. Zhao, Q. Dong, A strong convergence theorem by a relaxed extragradient method for a general system of variational inequalities and strict pseudo-contractions, Nonlinear Funct. Anal. Appl., 16 (2011), 465-479.


• [30] L. C. Zhao, S. S. Chang , Strong convergence theorems for equilibrium problems and fixed point problems of strict pseudo-contraction mappings, J. Nonlinear Sci. Appl., 2 (2009), 78-91.