WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

doi:10.1016/S0252-9602(12)60212-4

Abstract

Abstract:

In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.

Key words: weak convergence, relatively nonexpansive mapping, equilibrium problem, variational inequality

CLC Number:

47H09

CAI Gang, BU Shang-Quan. WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2013, 33(1): 303-320.

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References

[1] Cioranescu I. Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems. Dordrecht: Kluwer, 1990

[2] Takahashi S, Takahashi W. Strong convergence theorem for a generalized equilibrium problem and a nonexpansive mapping in a Hilbert space. Nonlinear Anal, 2008, 69: 1025–1033

[3] Peng J W, Yao J C. Ishikawa iterative algorithms for a generalized equilibrium problem and fixed point problems of a pseudo-contraction mapping. J Global Optim, 2010, 46: 331–345

[4] Zeng L C, Wu S Y, Yao J C. Generalized KKM theorem with applications to generalized minimax inequal-ities and generalized equilibrium problems. Taiwanese J Math, 2006, 10: 1497–1514

[5] Peng J W, Yao J C. A new hybrid-extragradient method for generalized mixed equilibrium problems and fixed point problems and variational inequality problems. Taiwanese J Math, 2008, 12: 1401–1433

[6] Peng J W, Yao J C. A new extragradient method for mixed equilibrium problems, fixed point problems and variational inequality problems. Math Comput Modelling, 2009, 49: 1816–1828

[7] Ceng L C, Ansari Q H, Yao J C. Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J Global Optim, 2009, 43: 487–502

[8] Ceng L C, et al. An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J Comput Appl Math, 2009, 223: 967–974

[9] Qin X, Cho S Y, Kang S M. Strong convergence of shrinking projection methods for quasi--nonexpansive mappings and equilibrium problems. J Comput Appl Math, 2010, 234: 750–760

[10] Qin X, Cho Y J, Kang S M. Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces. J Comput Appl Math, 2009, 225: 20–30

[11] Qin X, et al. Convergence of a modified Halpern-type iteration algorithm for quasi--nonexpansive map-pings. Appl Math Lett, 2009, 22: 1051–1055

[12] Chang S S, Joseph Lee H W, Chan C K. A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization. Nonlinear Anal, 2009, 70: 3307–3319

[13] Takahashi W, Zembayashi K. Strong and weak convergence theorems for equilibrium problems and rela-tively nonexpansive mappings in Banach spaces. Nonlinear Anal, 2009, 70: 45–57

[14] Wattanawitoon K, Kumam P. Strong convergence theorems by a new hybrid projection algorithm for fixed point problems and equilibrium problems of two relatively quasi-nonexpansive mappings. Nonlinear Anal, 2009, 3: 11–12

[15] Zegeye H, Shahzad N. Strong convergence theorems for monotone mappings and relatively weak nonex-pansive mappings. Nonlinear Anal, 2009, 70: 2707–2716

[16] Wei L, Cho Y J, Zhou H. A strong convergence theorem for common fixed points of two relatively nonex-pansive mappings and its applications. J Appl Math Comput, 2009, 29: 95–103

[17] Su Y, Wang Z, Xu H K. Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal, 2009, 71: 5616–5628

[18] Kimura Y, Takahashi W. On a hybrid method for a family of relatively nonexpansive mappings in a Banach space. J Math Anal Appl, 2009, 357: 356–363

[19] Jaiboon C, Kumam P, Humphries U W. Weak convergence theorem by an extragradient method for variational inequality, equilibrium and fixed point problems. Bull Malays Math Sci Soc, 2009, 32: 173–185

[20] Liou Y C. Shrinking projection method of proximal-type for a generalied equilibrium problem, a maximal monotone operator and a pair of relatively nonexpansive mappings. Taiwanese J Math, 2010, 14: 517–540

[21] Chang S S. Shrinking projection method for solving generalized equilibrium problem, variational inequality and common fixed point in Banach spaces with applications. Science in China Series A, to appear

[22] Zegeye H, Ofoedu E U, Shahzad N. Convergence theorems for equilibrium problem, variotional inequality problem and countably infinite relatively quasi-nonexpansive mappings. Appl Math Comput, 2010, 216: 3439–3449

[23] Martinet B. Regularisation dinequations variationnelles par approximations successives. Rev Franc Autom Inform Rech Oper, 1970, 4: 154–159

[24] Kohsaka F, Takahashi W. Strong convergence of an iterative sequence for maximal monotone operators in a Banach space. Abstr Appl Anal, 2004, 3: 239–249

[25] Alber Y I. Metric and generalized projection operators in Banach spaces: Properties and applications//Kartsatos A G, ed. Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. New York: Marcel Dekker, 1996: 15–50

[26] Kamimura S, Takahashi W. Strong convergence of a proximal-type algorithm in a Banach space. SIAM J Optim, 2002, 13: 938–945

[27] Butanriu D, Reich S, ZasIavski A J. Weak convergence of orbits of nonlinear operators in reflexive Banach spaces. Numer Funct Anal Optim, 2003, 24: 489–508

[28] Takahashi Y, Hashimoto K, Kato M. On sharp uniform convexity, smoothness, and strong type, cotype inequalities. J Nonlinear Convex Anal, 2002, 3: 267–281

[29] Xu H K. Inequalities in Banach spaces with applications. Nonlinear Anal, 1991, 16: 1127–1138

[30] Cho Y J, Zhou H, Guo G. Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput Math Appl, 2004, 47: 707–717

[31] Rockfellar R T. Monotone operators and the proximal point algorithm. SIAM J Control Optim, 1976, 14: 877–898

[32] Plubtieng S, Ungchittrakool K. Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. Nonlinear Anal, 2010, 72: 2896–2908

[33] Liu M, Zhang S S. A new iterative method for finding common solutions of generalized equilibrium prob-lem, fixed point problem of infinite k-strict pseudo-contractive mappings, and quasi-variational inclusion problem. Acta Mathematica Scientia, 2012, 32B(2): 499–519

[34] Jankaew E, Plubtieng S, Tepphun A. Viscosity iterative methods for common fixed points of two nonex-pansive mappings without commutativity assumption in Hilbert spaces. Acta Mathematica Scientia, 2011, 31B(2): 716–726

[35] Kim J K, Cho S Y, Qin X. Some results on generalized equilibrium problems involving strictly pseudocon-tractive mappings. Acta Mathematica Scientia, 2011, 31B(5): 2041–2057

[36] Gu F. Necessary and sufficient condition of the strong convergence for two finite families of uniformly L-Lipschitzian mappings. Acta Mathematica Scientia, 2011, 31B(5): 2058–2066