Abstract
A new iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, the set of common fixed points of two countable families of weak relatively nonexpansive mappings and the set of zeros of a maximal monotone operator in Banach spaces. The results obtained in this paper generalize and improve upon some existing results in recent literature.
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References
Alber Ya.: Metric and generalized projection operators in Banach spaces: Properties and applications. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp. 15–50. Dekker, New York (1996)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Butanriu D., Reich S., Zaslavski A.J.: Asymtotic behavior of relatively nonexpansive operators in Banach spaces. J. Appl. Anal. 7, 151–174 (2001)
Butanriu D., Reich S., Zaslavski A.J.: Weakly convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer. Funct. Anal. Optim. 24, 489–508 (2003)
Ceng L.C., Yao J.C.: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 214, 186–201 (2008)
Cho Y.J., Zhou H.Y., Guo G.: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 47, 707–717 (2004)
Kamimura S., Takahashi W.: Strong convergence of proximal-type algorithm in a Banach space. SIAM J. Optim. 13, 938–945 (2002)
Kamimura S., Kohsaka F., Takahashi W.: Weak and strong convergence theorem for maximal monotone operators in a Banach space. Set-Valued Anal. 12, 417–429 (2004)
Kohsaka F., Takahashi W.: Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr. Appl. Anal. 3, 239–249 (2004)
Kohsaka F., Takahashi W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19(2), 824–835 (2008)
Pascali D., Sburlan S.: Nonlinear Mappings of Monotone Type, Editura Academiae. Bucaresti, Romania (1978)
Reich S.: A weak convergence theorem for the alternating method with Bergman distance. In: Kartsatos, A.G. (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes in Pure and Appl. Math., vol. 178, pp. 313–318. Dekker, New York (1996)
Rockafellar R.T.: On the maximality of sums of nonlinear monotone operators. Trans. Am. Math. Soc. 149, 75–88 (1970)
Rockafellar R.T.: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 14, 877–898 (1976)
Su Y.F., Wang Z.M., Xu H.K.: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616–5628 (2009)
Su Y.F., Xu H.K., Zhang X.: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 73, 3890–3906 (2010)
Takahashi W.: Nonlinear Functional Analysis. Kindikagaku, Tokyo (1988) (in Japanese)
Takahashi S., Takahashi W.: Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal. 69, 1025–1033 (2008)
Takahashi W., Zembayashi K.: Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces. Nonlinear. Anal. 70, 45–57 (2009)
Wang Y.Q., Zeng L.C.: Hybrid projection method for generalized mixed equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces. Appl. Math. Mech. Engl. Ed. 32(2), 251–264 (2011)
Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)
Zegeye H., Shahzad N.: Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings. Nonlinear Anal. 70, 2707–2716 (2009)
Zhang S.S.: Generalized mixed equilibrium problem in Banach spaces. Appl. Math. Mech. Engl. Ed. 30(9), 1105–1112 (2009)
Acknowledgments
This work was done during the first and third authors’ visit to the Research Center for Nonlinear Analysis and Discrete Mathematics at the National Sun Yat-sen University. The second author extends his appreciation to the Deanship of Scientific Research at King Saud University for funding the work through a visiting professorship program.
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Wang, Y., Xu, HK. & Yin, X. Strong convergence theorems for generalized equilibrium, variational inequalities and nonlinear operators. Arab. J. Math. 1, 549–568 (2012). https://doi.org/10.1007/s40065-012-0032-3
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DOI: https://doi.org/10.1007/s40065-012-0032-3