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A viscosity extragradient method for an equilibrium problem and fixed point problem in Hilbert space. (English) Zbl 1421.47003

Let \(H\) be a real Hilbert space and \(C\) be its nonempty, closed and convex subset. Let \(T:C\to C\) be a nonexpansive mapping and \(f:C\times C\to \mathbb{R}\) be a function satisfying the following conditions: (1) \(f\) is pseudomonotone and \(f(x,x)=0\) for each \(x\in C\); (2) \(f\) is Lipschitz-type continuous on \(C\); (3) \(f(x,\cdot)\) is convex and subdifferentiable, for each \(x\in C\); (4) \(f(x,y)\) is jointly weakly continuous on \(C\times C\). Assume that \(A=\operatorname{Fix}(T)\cap EP(f)\neq\emptyset\), where \(\operatorname{Fix}(T)\) is the set of fixed points of \(T\) and \(EP(f)=\{z\in C:f(z,y)\geq0\text{ for all }y\in C\}\) is the set of solutions of the equilibrium problem of \(f\). The authors present an algorithm to find an element in \(A\). A numerical example illustrates the algorithm.

MSC:

47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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[1] Anh, P.N.: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 63, 271-283 (2013) · Zbl 1290.90084 · doi:10.1080/02331934.2011.607497
[2] Anh, P.N.: A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems. Bull. Malays. Math. Sci. 36, 107-116 (2013) · Zbl 1263.65066
[3] Anh, P.N.: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 154, 303-320 (2012) · Zbl 1270.90100 · doi:10.1007/s10957-012-0005-x
[4] Anh, P.N., An, L.T.H.: The subgradient extragradient method extended to equilibrium problems. Optimization 64, 225-248 (2015) · Zbl 1317.65149 · doi:10.1080/02331934.2012.745528
[5] Anh, P.N., Le Thi, H.A.: An Armijo-type method for pseudomonotone equilibrium problems and its applications. J. Glob. Optim. 57, 803-820 (2013) · Zbl 1285.65040 · doi:10.1007/s10898-012-9970-8
[6] Chang, S.-S., Lee, H.W.J., Chan, C.K.: A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal. Theory Methods Appl. 70, 3307-3319 (2009) · Zbl 1198.47082 · doi:10.1016/j.na.2008.04.035
[7] Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming using proximal like algorithms. Math. Prog. 78, 29-41 (1997) · doi:10.1016/S0025-5610(96)00071-8
[8] Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hiblert spaces. J. Nonlinear Convex Anal. 6, 117-136 (2005) · Zbl 1109.90079
[9] Dinh, B.V., Kim, D.S.: Projection algorithms for solving nonmonotone equilibrium problems in Hilbert space. J. Comput. Appl. Math. 302, 106-117 (2016) · Zbl 1334.90125 · doi:10.1016/j.cam.2016.01.054
[10] Goebel, K., Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings. Marcel Dekker, New York (1984) · Zbl 0537.46001
[11] Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197-217 (2016) · Zbl 1367.65089 · doi:10.1007/s11075-015-0092-5
[12] Kang, S.M., Cho, S.Y., Liu, Z.: Convergence of iterative sequences for generalized equilibrium problems involving inverse-strongly monotone mappings. J. Inequal. Appl. 2010, 827082 (2010) · Zbl 1187.47050
[13] Kassay, G., Reich, S., Sabach, S.: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 21, 1319-1344 (2011) · Zbl 1250.47064 · doi:10.1137/110820002
[14] Katchang, P., Kumam, P.: A new iterative algorithm of solution for equilibriumproblems, variational inequalities and fixed point problems in a Hilbert space. J. Appl. Math. Comput. 32, 19-38 (2010) · Zbl 1225.47100 · doi:10.1007/s12190-009-0230-0
[15] Maingé, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899-912 (2008) · Zbl 1156.90426 · doi:10.1007/s11228-008-0102-z
[16] Plubtieng, S., Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455-469 (2007) · Zbl 1127.47053 · doi:10.1016/j.jmaa.2007.02.044
[17] Qin, X., Cho, Y.J., Kang, S.M.: Viscosity approximation methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. 72, 99-112 (2010) · Zbl 1225.47106 · doi:10.1016/j.na.2009.06.042
[18] Qin, X., Shang, M., Su, Y.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. Nonlinear Anal. 69, 3897-3909 (2008) · Zbl 1170.47044 · doi:10.1016/j.na.2007.10.025
[19] Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225-240 (2012) · Zbl 1293.47065 · doi:10.1090/conm/568/11285
[20] Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359-370 (2007) · Zbl 1147.47052 · doi:10.1007/s10957-007-9187-z
[21] Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506-515 (2007) · Zbl 1122.47056 · doi:10.1016/j.jmaa.2006.08.036
[22] 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) · Zbl 1142.47350 · doi:10.1016/j.na.2008.02.042
[23] Tran, D.Q., Dung, M.L., Nguyen, V.H.: Extragradient algorithms extended to equilibrium problem. Optimization 57, 749-776 (2008) · Zbl 1152.90564 · doi:10.1080/02331930601122876
[24] Vuong, P.T., Strodiot, J.J., Nguyen, V.H.: Extragradient methods and linear algorithms for solving Ky Fan inequalities and fixed point problems. J. Optim. Theory Appl. 155, 605-627 (2012) · Zbl 1273.90207 · doi:10.1007/s10957-012-0085-7
[25] Wen, D.J., Chen, Y.A.: General iterative methods for generalized equilibrium problems and fixed point problems of k-strict pseudo-contractions. Fixed Point Theory Appl. 2012, 125 (2012) · Zbl 1475.47080 · doi:10.1186/1687-1812-2012-125
[26] Xu, H.K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Aust. Math. Soc. 65, 109-113 (2002) · Zbl 1030.47036 · doi:10.1017/S0004972700020116
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