×

New iterative approximation methods for a countable family of nonexpansive mappings in Banach spaces. (English) Zbl 1207.65071

Summary: We introduce new general iterative approximation methods for finding a common fixed point of a countable family of nonexpansive mappings. Strong convergence theorems are established in the framework of reflexive Banach spaces which admit a weakly continuous duality mapping. Finally, we apply our results to solve the the equilibrium problems and the problem of finding a zero of an accretive operator. The results presented in this paper mainly improve on the corresponding results reported by many others.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space.Nonlinear Analysis: Theory, Methods & Applications 2007,67(8):2350-2360. 10.1016/j.na.2006.08.032 · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032
[2] Bauschke HH: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space.Journal of Mathematical Analysis and Applications 1996,202(1):150-159. 10.1006/jmaa.1996.0308 · Zbl 0956.47024 · doi:10.1006/jmaa.1996.0308
[3] Shang, M.; Su, Y.; Qin, X., Strong convergence theorems for a finite family of nonexpansive mappings, No. 2007, 9 (2007) · Zbl 1155.47314
[4] Shimoji K, Takahashi W: Strong convergence to common fixed points of infinite nonexpansive mappings and applications.Taiwanese Journal of Mathematics 2001,5(2):387-404. · Zbl 0993.47037
[5] Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems.SIAM Review 1996,38(3):367-426. 10.1137/S0036144593251710 · Zbl 0865.47039 · doi:10.1137/S0036144593251710
[6] Combettes PL: Foundations of set theoretic estimation.Proceedings of the IEEE 1993,81(2):182-208. · doi:10.1109/5.214546
[7] Youla, DC; Stark, H. (ed.), Mathematical theory of image restoration by the method of convex projections, 29-77 (1987), Orlando, Fla, USA
[8] Iusem AN, De Pierro AR: On the convergence of Han’s method for convex programming with quadratic objective.Mathematical Programming. Series B 1991,52(2):265-284. · Zbl 0744.90066 · doi:10.1007/BF01582891
[9] Browder FE: Fixed-point theorems for noncompact mappings in Hilbert space.Proceedings of the National Academy of Sciences of the United States of America 1965, 53: 1272-1276. 10.1073/pnas.53.6.1272 · Zbl 0125.35801 · doi:10.1073/pnas.53.6.1272
[10] Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces.Journal of Mathematical Analysis and Applications 1980,75(1):287-292. 10.1016/0022-247X(80)90323-6 · Zbl 0437.47047 · doi:10.1016/0022-247X(80)90323-6
[11] Xu H-K: Strong convergence of an iterative method for nonexpansive and accretive operators.Journal of Mathematical Analysis and Applications 2006,314(2):631-643. 10.1016/j.jmaa.2005.04.082 · Zbl 1086.47060 · doi:10.1016/j.jmaa.2005.04.082
[12] Deutsch F, Yamada I: Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings.Numerical Functional Analysis and Optimization 1998,19(1-2):33-56. · Zbl 0913.47048 · doi:10.1080/01630569808816813
[13] Xu H-K: An iterative approach to quadratic optimization.Journal of Optimization Theory and Applications 2003,116(3):659-678. 10.1023/A:1023073621589 · Zbl 1043.90063 · doi:10.1023/A:1023073621589
[14] Xu H-K: Iterative algorithms for nonlinear operators.Journal of the London Mathematical Society. Second Series 2002,66(1):240-256. 10.1112/S0024610702003332 · Zbl 1013.47032 · doi:10.1112/S0024610702003332
[15] Moudafi A: Viscosity approximation methods for fixed-points problems.Journal of Mathematical Analysis and Applications 2000,241(1):46-55. 10.1006/jmaa.1999.6615 · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615
[16] Xu H-K: Viscosity approximation methods for nonexpansive mappings.Journal of Mathematical Analysis and Applications 2004,298(1):279-291. 10.1016/j.jmaa.2004.04.059 · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059
[17] Marino G, Xu H-K: A general iterative method for nonexpansive mappings in Hilbert spaces.Journal of Mathematical Analysis and Applications 2006,318(1):43-52. 10.1016/j.jmaa.2005.05.028 · Zbl 1095.47038 · doi:10.1016/j.jmaa.2005.05.028
[18] Halpern B: Fixed points of nonexpanding maps.Bulletin of the American Mathematical Society 1967, 73: 957-961. 10.1090/S0002-9904-1967-11864-0 · Zbl 0177.19101 · doi:10.1090/S0002-9904-1967-11864-0
[19] Song Y, Zheng Y: Strong convergence of iteration algorithms for a countable family of nonexpansive mappings.Nonlinear Analysis: Theory, Methods & Applications 2009,71(7-8):3072-3082. 10.1016/j.na.2009.01.219 · Zbl 1222.47119 · doi:10.1016/j.na.2009.01.219
[20] Jung JS: Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces.Journal of Mathematical Analysis and Applications 2005,302(2):509-520. 10.1016/j.jmaa.2004.08.022 · Zbl 1062.47069 · doi:10.1016/j.jmaa.2004.08.022
[21] O’Hara JG, Pillay P, Xu H-K: Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces.Nonlinear Analysis: Theory, Methods & Applications 2003,54(8):1417-1426. 10.1016/S0362-546X(03)00193-7 · Zbl 1052.47049 · doi:10.1016/S0362-546X(03)00193-7
[22] O’Hara JG, Pillay P, Xu H-K: Iterative approaches to convex feasibility problems in Banach spaces.Nonlinear Analysis: Theory, Methods & Applications 2006,64(9):2022-2042. 10.1016/j.na.2005.07.036 · Zbl 1139.47056 · doi:10.1016/j.na.2005.07.036
[23] Wangkeeree, R., An extragradient approximation method for equilibrium problems and fixed point problems of a countable family of nonexpansive mappings, No. 2008, 17 (2008) · Zbl 1170.47051
[24] Wangkeeree, R.; Kamraksa, U., A general iterative method for solving the variational inequality problem and fixed point problem of an infinite family of nonexpansive mappings in Hilbert spaces, No. 2009, 23 (2009) · Zbl 1168.47054
[25] Wangkeeree R, Petrot N, Wangkeeree R: The general iterative methods for nonexpansive mappings in Banach spaces.Journal of Global Optimization. In press · Zbl 1471.65041
[26] Takahashi W: Nonlinear Functional Analysis: Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama, Japan; 2000:iv+276. · Zbl 0997.47002
[27] Browder FE: Convergence theorems for sequences of nonlinear operators in Banach spaces.Mathematische Zeitschrift 1967, 100: 201-225. 10.1007/BF01109805 · Zbl 0149.36301 · doi:10.1007/BF01109805
[28] Lim T-C, Xu H-K: Fixed point theorems for asymptotically nonexpansive mappings.Nonlinear Analysis: Theory, Methods & Applications 1994,22(11):1345-1355. 10.1016/0362-546X(94)90116-3 · Zbl 0812.47058 · doi:10.1016/0362-546X(94)90116-3
[29] Peng J-W, Yao J-C: A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings.Nonlinear Analysis: Theory, Methods & Applications 2009,71(12):6001-6010. 10.1016/j.na.2009.05.028 · Zbl 1178.47047 · doi:10.1016/j.na.2009.05.028
[30] Eshita K, Takahashi W: Approximating zero points of accretive operators in general Banach spaces.JP Journal of Fixed Point Theory and Applications 2007,2(2):105-116. · Zbl 1139.47044
[31] Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems.The Mathematics Student 1994,63(1-4):123-145. · Zbl 0888.49007
[32] Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces.Journal of Nonlinear and Convex Analysis 2005,6(1):117-136. · Zbl 1109.90079
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.