# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. (English) Zbl 1166.65026
The author discusses the convergence of an explicit iterative schemes involving a sequence of nonexpansive mappings $\{T_n\}$ on a real Banach space (with some properties), and also a contraction $f$. A general framework is developed to prove the strong convergence of the iterative schemes to the fixed point of a nonexpansive mapping $T$, related to the sequence $\{T_n\}$. By specifying the sequence $\{T_n\}$, the author recovers and extends some known convergence theorems. In the iterative schemes, the contraction $f$ can be replaced by the Meir-Keeler contraction [see {\it A. Meir} and {\it E. Keeler}, J. Math. Anal. Appl. 28, 326--329 (1969; Zbl 0194.44904)]. Seven examples of iterative schemes from the literature which are generalized by the iterative schemes proposed by the author, are analyzed in detail.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties
Full Text:
##### References:
 [1] Aoyoma, K.; Kimura, Y.; Takahashi, W.; Toyoda, M.: Approximation of common fixed point of a countable family of nonexpansive mappings, Nonlinear anal. 67, No. 8, 2350-2360 (2007) · Zbl 1130.47045 · doi:10.1016/j.na.2006.08.032 [2] Bauschke, H. H.; Borwein, J. M.: On projection algorithms for solving convex feasibility problems, SIAM rev. 38, No. 3, 367-426 (1996) · Zbl 0865.47039 · doi:10.1137/S0036144593251710 [3] Bruck, R. E.: Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. amer. Math. soc. 179, 251-262 (1973) · Zbl 0265.47043 · doi:10.2307/1996502 [4] Chen, J.; Zhang, L.; Fan, T.: Viscosity approximation methods for nonexpansive mappings and monotone mappings, J. math. Anal. appl. 334, No. 2, 1450-1461 (2007) · Zbl 1137.47307 · doi:10.1016/j.jmaa.2006.12.088 [5] Combettes, P.; Histoaga, S.: Equilibrium programming in Hilbert spaces, J. nonlinear convex anal. 6, No. 1, 117-136 (2005) · Zbl 1109.90079 [6] Flåm, S.; Antipin, A.: Equilibrium programming using proximal-like algorithms, Math. program. 77, 29-41 (1997) · Zbl 0890.90150 · doi:10.1007/BF02614504 [7] Goebel, K.; Reich, S.: Uniform convexity, hyperbolic geometry, and nonexpansive mappings, (1984) · Zbl 0537.46001 [8] Gossez, J. -P.; Dozo, E. L.: Some geometric properties related to the fixed point theory for nonexpansive mappings, Pacific J. Math. 40, No. 3, 565-573 (1972) · Zbl 0223.47025 [9] Kim, T. -H.; Xu, H. -K.: Strong convergence of modified Mann iterations, Nonlinear anal. 61, No. 1 -- 2, 51-60 (2005) · Zbl 1091.47055 [10] Kimura, Y.; Takahashi, W.; Toyoda, M.: Convergence to common fixed points of a finite family of nonexpansive mappings, Arch. math. 84, 350-363 (2005) · Zbl 1086.47051 · doi:10.1007/s00013-004-1175-z [11] G. Lopez, V. Martin, H.-K. Xu, Perturbation techniques for nonexpansive mappings with applications, Nonlinear Anal. Real World Appl., in press, available online 4 May 2008 [12] Meir, A.; Keeler, E.: A theorem on contraction mappings, J. math. Anal. appl. 28, 326-329 (1969) · Zbl 0194.44904 · doi:10.1016/0022-247X(69)90031-6 [13] Moudafi, A.: Viscosity approximation methods for fixed-points problems, J. math. Anal. appl. 241, 46-55 (2000) · Zbl 0957.47039 · doi:10.1006/jmaa.1999.6615 [14] Plubtieng, S.; Punpaeng, R.: A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. math. Anal. appl. 336, No. 1, 455-469 (2007) · Zbl 1127.47053 · doi:10.1016/j.jmaa.2007.02.044 [15] Song, Y.; Chen, R.: Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. math. Comput. 180, 275-287 (2006) · Zbl 1139.47050 · doi:10.1016/j.amc.2005.12.013 [16] Song, Y.; Chen, R.: Viscosity approximation methods for nonexpansive nonself-mappings, J. math. Anal. appl. 321, No. 1, 316-326 (2006) · Zbl 1103.47053 · doi:10.1016/j.jmaa.2005.07.025 [17] Suzuki, T.: Moudafi’s viscosity approximations with Meir -- Keeler contractions, J. math. Anal. appl. 325, No. 1, 342-352 (2007) · Zbl 1111.47059 · doi:10.1016/j.jmaa.2006.01.080 [18] Takahashi, W.; Tamura, T.; Toyoda, M.: Approximation of common fixed points of a family of finite nonexpansive mappings in Banach spaces, Sci. math. Jpn. 56, No. 3, 475-480 (2002) · Zbl 1026.47042 [19] Xu, H. -K.: Viscosity approximation methods for nonexpansive mappings, J. math. Anal. appl. 298, No. 1, 279-291 (2004) · Zbl 1061.47060 · doi:10.1016/j.jmaa.2004.04.059 [20] Xu, H. -K.: Strong convergence of an iterative method for nonexpansive and accretive operators, J. math. Anal. appl. 314, 631-643 (2006) · Zbl 1086.47060 · doi:10.1016/j.jmaa.2005.04.082