Fixed point theory for nonlinear mappings in Banach spaces and applications
Tóm tắt
The purpose of this research is to study a finite family of the set of solutions of variational inequality problems and to prove a convergence theorem for the set of such problems and the sets of fixed points of nonexpansive and strictly pseudo-contractive mappings in a uniformly convex and 2-uniformly smooth Banach space. We also prove a fixed point theorem for finite families of nonexpansive and strictly pseudo-contractive mappings in the last section.
Tài liệu tham khảo
Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3
Kopecka E, Reich S: Nonexpansive retracts in Banach spaces. Banach Cent. Publ. 2007, 77: 161–174.
Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006: Article ID 35390 10.1155/FPTA/2006/35390
Ceng LC, Wang CY, Yao JC: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008, 67: 375–390. 10.1007/s00186-007-0207-4
Yao Y, Liou YC, Kang SM: Approach to common elements of variational inequality problems and fixed point problems via a relaxed extragradient method. Comput. Appl. Math. 2010, 59: 3472–3480. 10.1016/j.camwa.2010.03.036
Qin X, Kang SM: Convergence theorems on an iterative method for variational inequality problems and fixed point problems. Bull. Malays. Math. Soc. 2010, 33: 155–167.
Censor Y, Gibali A, Reich S: The subgradient extragradient method for solving variational inequalities in Hilbert space. J. Optim. Theory Appl. 2011, 148: 318–335. 10.1007/s10957-010-9757-3
Kassay G, Reich S, Sabach S: Iterative methods for solving systems of variational inequalities in reflexive Banach spaces. SIAM J. Optim. 2011, 21: 1319–1344. 10.1137/110820002
Yao Y, Cho YJ, Chen R: An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. Nonlinear Anal. 2009, 71: 3363–3373. 10.1016/j.na.2009.01.236
Qin X, Shang M, Su Y: Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems. Math. Comput. Model. 2008, 48: 1033–1046. 10.1016/j.mcm.2007.12.008
Kangtunyakarn A: A new mapping for finding a common element of the sets of fixed points of two finite families of nonexpansive and strictly pseudo-contractive mappings and two sets of variational inequalities in uniformly convex and 2-smooth Banach spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 157
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2
Zhou H: Convergence theorems for κ -strict pseudocontractions in 2-uniformly smooth Banach spaces. Nonlinear Anal. 2008, 69: 3160–3173. 10.1016/j.na.2007.09.009
Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589