Abstract
In this paper, an iterative algorithm is investigated for a common solution problem. Strong convergence of the algorithm is obtained in the framework of real Hilbert spaces. Applications are also provided to support the main results.
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References
Bauschke HH, Borwein JM (1996) On projection algorithms for solving convex feasibility problems. SIAM Rev 38:367–426
Blum E, Oettli W (1994) From optimization and variational inequalities to equilibrium problems. Math Stud 63:123–145
Chang SS, Lee HWJ, Chan CK (2009) A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal 70:3307–3319
Chen JH (2013) Iterations for equilibrium and fixed point problems. J Nonlinear Funct Anal 2013(4)
Cheng P, Wu H (2013) On asymptotically strict pseudocontractions and equilibrium problems. J Inequal Appl 2013(251)
Cho SY, Kang SM (2011) Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett 24:224–228
Cho SY, Kang SM (2012) Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci 32:1607–1618
Cho SY, Li W, Kang SM (2013) Convergence analysis of an iterative algorithm for monotone operators. J Inequal Appl 2013:199
Cho SY, Qin X (2014) On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems. Appl Math Comput 235:430–438
Combettes PL (1996) The convex feasibility problem in image recovery. Adv Imag Electron Phys 95:155–270
Hao Y (2010) On variational inclusion and common fixed point problems in Hilbert spaces with applications. Appl Math Comput 217:3000–3010
He RH (2012) Coincidence theorem and existence theorems of solutions for a system of Ky Fan type minimax inequalities in FC-spaces. Adv Fixed Point Theory 2:47–57
Khanh PQ, Luu LM (2004) On the existence of solutions to vector quasivariational inequalities and quasicomplementarity problems with applications to traffic network equilibria. J Optim Theory Appl 123:533–548
Lions PL, Mercier B (1979) Splitting algorithms for the sum of two nonlinear operators. SIAM J Numer Anal 16:964–979
Liu LS (1995) Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 194:114–125
Liu M, Li X, Pu D (2013) A tri-dimensional filter SQP algorithm for variational inequality problems. Comput Math Appl 32:549–562
Park S (2013) A review of the KKM theory on \(\phi _A\)-spaces or GFC-spaces. Adv Fixed Point Theor 3:355–382
Qin X, Cho SY, Kang SM (2010) Some results on variational inequalities and generalized equilibrium problems with applications. Comput Appl Math 29:393–421
Qin X, Chang SS, Cho YJ (2010) Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal 11:2963–2972
Rockafellar RT (1970) On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 149:75–88
Rodjanadid B, Sompong S (2013) A new iterative method for solving a system of generalized equilibrium problems, generalized mixed equilibrium problems and common fixed point problems in Hilbert spaces. Adv Fixed Point Theory 3:675–705
Shimoji K, Takahashi W (2001) Strong convergence to common fixed points of infinite nonexpansive mappings and applications. Taiwanese J Math 5:387–404
Suzuki T (2005) Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochne integrals. J Math Anal Appl 305:227–239
Takahashi S, Takahashi W (2007) Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J Math Anal Appl 331:506–515
Wang L (2014) A regularization method for treating zero points of the sum of two monotone operators. Fixed Point Theory Appl 2014(75)
Wu C, Liu A (2012) Strong convergence of a hybrid projection iterative algorithm for common solutions of operator equations and of inclusion problems. Fixed Point Theory Appl 2012(90)
Zegeye H, Shahzad N (2012) Strong convergence theorem for a common point of solution of variational inequality and fixed point problem. Adv Fixed Point Theory 2:374–397
Acknowledgments
The author is Grateful to the anonymous reviewers for useful suggestions which improved the contents of the article. This work is supported by the Fundamental Research Funds for the Central Universities (Grant No. 2014ZD44).
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Communicated by Carlos Conca.
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Wang, S. Strong convergence of a regularization algorithm for common solutions with applications. Comp. Appl. Math. 35, 153–169 (2016). https://doi.org/10.1007/s40314-014-0187-y
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DOI: https://doi.org/10.1007/s40314-014-0187-y