Strong convergence theorems for fixed point problems of a nonexpansive semigroup in a Banach space. (English) Zbl 1476.47082
Summary: In this paper, we study the implicit and explicit viscosity iteration schemes for a nonexpansive semigroup in a reflexive, strictly convex and uniformly smooth Banach space which satisfies Opial’s condition. Our results improve and generalize the corresponding results given by Y.-H. Yao et al. [Fixed Point Theory Appl. 2013, Paper No. 31, 11 p. (2013; Zbl 1281.47062)] and many others.
MSC:
| 47J25 | Iterative procedures involving nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
| 47H20 | Semigroups of nonlinear operators |
Keywords:
nonexpansive semigroup; fixed point; reflexive and strictly convex Banach space; uniformly smooth Banach space; Opial condition; sunny nonexpansive retractionCitations:
Zbl 1281.47062References:
| [1] | Aleyner A, Censor Y: Best approximation to common fixed points of a semigroup of nonexpansive operators.J. Nonlinear Convex Anal. 2005, 6:137-151. · Zbl 1071.41031 |
| [2] | Sunthrayuth P, Kumam P: A general iterative algorithm for the solution of variational inequalities for a nonexpansive semigroup in Banach spaces.J. Nonlinear Anal. Optim. 2010, 1:139-150. · Zbl 1413.47141 |
| [3] | Yao, Y, Liou, Y-C, Yao, J-C: Algorithms for finding minimum norm solution of equilibrium and fixed point problems for nonexpansive semigroups in Hilbert spaces. J. Nonlinear Convex Anal. (in press) · Zbl 1347.47042 |
| [4] | Yang, P.; Yao, Y.; Liou, Y-C; Chen, R., Hybrid algorithms of nonexpansive semigroups for variational inequalities (2012) · Zbl 1251.49014 |
| [5] | Yao, Y.; Cho, YJ; Liou, Y-C, Hierarchical convergence of an implicit double-net algorithm for nonexpansive semigroups and variational inequalities, No. 2011 (2011) · Zbl 1275.49018 |
| [6] | Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach space.Appl. Math. Lett. 2007, 20:751-757. · Zbl 1161.47049 · doi:10.1016/j.aml.2006.09.003 |
| [7] | Yao, Y.; Kang, JI; Cho, YJ; Liou, Y-C, Approximation of fixed points for nonexpansive semigroup in Hilbert spaces (2013) |
| [8] | Gossez J-P, Lami Dozo E: Some geometric properties related to the fixed point theory for nonexpansive mappings.Pac. J. Math. 1972, 40:565-573. · Zbl 0223.47025 · doi:10.2140/pjm.1972.40.565 |
| [9] | Goebel K, Reich S: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York; 1984. · Zbl 0537.46001 |
| [10] | Kopecka E, Reich S: Nonexpansive retracts in Banach spaces.Banach Cent. Publ. 2007, 77:161-174. · Zbl 1125.46019 · doi:10.4064/bc77-0-12 |
| [11] | Reich S: Asymptotic behavior of contractions in Banach spaces.J. Math. Anal. Appl. 1973, 44:57-70. · Zbl 0275.47034 · doi:10.1016/0022-247X(73)90024-3 |
| [12] | Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions.Topol. Methods Nonlinear Anal. 1993, 2:333-342. · Zbl 0815.47068 |
| [13] | Jung JS: Iterative approach to common fixed points of nonexpansive mappings in Banach spaces.J. Math. Anal. Appl. 2005, 302:509-520. · Zbl 1062.47069 · doi:10.1016/j.jmaa.2004.08.022 |
| [14] | Suzuki T: Strong convergence of Krasnoselskii and Mann’s sequences for one-parameter nonexpansive semigroup without Bochner integrals.J. Math. Anal. Appl. 2005, 305:227-239. · Zbl 1068.47085 · doi:10.1016/j.jmaa.2004.11.017 |
| [15] | Xu HK: An iterative approach to quadratic optimization.J. Optim. Theory Appl. 2003, 116:659-678. · Zbl 1043.90063 · doi:10.1023/A:1023073621589 |
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