Strong convergence theorems for a finite family of total asymptotically strict pseudocontractive semigroups in Banach spaces. (English) Zbl 1476.47037
Summary: The purpose of this paper is to introduce the concepts of total asymptotically strictly pseudocontractive semigroup, asymptotically strictly pseudocontractive semigroup etc., and to prove some strong convergence theorems of the explicit iteration process for these kinds of semigroups in arbitrary Banach spaces. The results presented in the paper extend and improve some recent results announced in the current literature.
MSC:
| 47H20 | Semigroups of nonlinear operators |
| 47H09 | Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. |
Keywords:
total asymptotically strict pseudocontractive semigroup; asymptotically strict pseudocontractive semigroup; asymptotically demicontractive semigroup; fixed point; normalized duality mappingReferences:
| [1] | Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive mappings in Hilbert spaces.Nonlinear Anal. 1998,34(1):87-99. · Zbl 0935.47039 · doi:10.1016/S0362-546X(97)00682-2 |
| [2] | Suzuki T: On strong convergence to a common fixed point of nonexpansive semigroups in Hilbert spaces.Proc. Am. Math. Soc. 2003,131(7):2133-2136. · Zbl 1031.47038 · doi:10.1090/S0002-9939-02-06844-2 |
| [3] | Xu HK: Strong convergence theorem for contraction semigroups in Banach spaces.Bull. Aust. Math. Soc. 2005,72(3):371-379. · Zbl 1095.47016 · doi:10.1017/S000497270003519X |
| [4] | Aleyner A, Reich S: An explicit construction of sunny nonexpansive retractions in Banach spaces.Fixed Point Theory Appl. 2005, 3:295-305. · Zbl 1096.47055 |
| [5] | Zhang SS, Yang L, Liu JA: Strong convergence theorem for nonexpansive semigroups in Banach spaces.Appl. Math. Mech. 2007,28(10):1287-1297. · Zbl 1231.47060 · doi:10.1007/s10483-007-1002-x |
| [6] | Li S, Li LH, Su F: General iteration methods for a one-parameter nonexpansive semigroups in Hilbert spaces.Nonlinear Anal. 2009,70(9):3065-3071. · Zbl 1177.47075 · doi:10.1016/j.na.2008.04.007 |
| [7] | Zhang SS, Yang L, Lee HWJ, Chan CK: Strong convergence theorem for nonexpansive semigroups in Hilbert spaces.Acta Math. Sin. 2009,52(2):337-342. · Zbl 1199.47235 |
| [8] | Suzuki T: Fixed point property for nonexpansive mappings versus that for nonexpansive semigroups.Nonlinear Anal. 2009, 70:3358-3361. · Zbl 1221.47096 · doi:10.1016/j.na.2008.05.003 |
| [9] | Zhang SS: Convergence theorem of common fixed points for Lipschitzian pseudocontraction semigroups in Banach spaces.Appl. Math. Mech. 2009, 30:145-152. · Zbl 1181.47074 · doi:10.1007/s10483-009-0202-y |
| [10] | Chang SS, Chan CK, Lee HWJ, Yang L: A system of mixed equilibrium problems, fixed point problems of strictly pseudocontractive mappings and nonexpansive semigroups.Appl. Math. Comput. 2010,216(1):51-60. · Zbl 1191.65061 · doi:10.1016/j.amc.2009.12.060 |
| [11] | Chang, SS; Cho, YJ; Lee, HWJ; Chan, CK, Strong convergence theorems for Lipschitzian demicontraction semigroups in Banach spaces (2011) · Zbl 1215.47062 |
| [12] | Chang SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping equations in Banach spaces.J. Math. Anal. Appl. 1997, 216:94-111. · Zbl 0909.47049 · doi:10.1006/jmaa.1997.5661 |
| [13] | Lau AT-M, Zhang Y: Fixed point properties of semigroups of non-expansive mappings.J. Funct. Anal. 2008,254(10):2534-2554. · Zbl 1149.47046 · doi:10.1016/j.jfa.2008.02.006 |
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.
