# 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)
Strong convergence of an explicit iteration process for a totally asymptotically $I$-nonexpansive mapping in Banach spaces. (English) Zbl 1201.47064
Summary: We prove the strong convergence of an explicit iterative process to a common fixed point of a totally asymptotically $I$-nonexpansive mapping $T$ and a totally asymptotically nonexpansive mapping $I$, defined on a nonempty closed convex subset of a uniformly convex Banach space.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties
##### References:
 [1] Goebel, K.; Kirk, W. A.: A fixed point theorem for asymptotically nonexpansive mapping, Proc. amer. Math. soc. 35, 171-174 (1972) · Zbl 0256.47045 · doi:10.2307/2038462 [2] Bruck, B.; Kuczumow, T.; Reich, S.: Convergence of iterates of asymptotically nonexpansive mappings in Banach space with the uniform Opial property, Colloq. math. 65, No. 2, 169-179 (1993) · Zbl 0849.47030 [3] Kirk, W. A.: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math. 17, No. 2, 339-346 (1974) · Zbl 0286.47034 · doi:10.1007/BF02757136 [4] Bruck, R. E.: Asymptotic behavior of nonexpansive mappings, (1980) [5] Gornicki, J.: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly Banach space, Comment. math. Univ. carolin. 30, No. 2, 249-252 (1989) · Zbl 0686.47045 [6] Chidumi, Ch.: Geometric properties of Banach spaces and nonlinear iterations, Lecture notes math. 1965 (2009) [7] Alber, Ya.I.; Chidume, C. E.; Zegeye, H.: Approximating fixed points of total asymptotically nonexpansive mappings, Fixed point theory and appl. 2006 (2006) · Zbl 1105.47057 · doi:10.1155/FPTA/2006/10673 [8] Chidume, C. E.; Ofoedu, E. U.: Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, J. math. Anal. appl. 333, No. 1, 128-141 (2007) · Zbl 1127.47051 · doi:10.1016/j.jmaa.2006.09.023 [9] Chidume, C. E.; Ofoedu, E. U.: A new iteration process for approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, Internat. J. Math. math. Sci. 2009 (2009) · Zbl 1186.47059 · doi:10.1155/2009/615107 [10] Shahzad, N.: Generalized I-nonexpansive maps and best approximations in Banach spaces, Demonstratio math. 37, No. 3, 597-600 (2004) · Zbl 1095.41017 [11] Temir, S.: On the convergence theorems of implicit iteration process for a finite family of I-asymptotically nonexpansive mappings, J. comput. Appl. math. 225, 398-405 (2009) · Zbl 1162.47053 · doi:10.1016/j.cam.2008.07.049 [12] Mukhamedov, F.; Saburov, M.: Weak and strong convergence of an implicit iteration process for an asymptotically quasi-I-nonexpansive mapping in Banach space, Fixed point theory appl. 2010 (2010) · Zbl 1202.47080 · doi:10.1155/2010/719631 [13] Schu, J.: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. austral. Math. soc. 43, No. 1, 153-159 (1991) · Zbl 0709.47051 · doi:10.1017/S0004972700028884 [14] Tan, K. -K.; Xu, H. -K.: Approximating fixed points of nonexpansive mapping by the Ishikawa iteration process, J. math. Anal. appl. 178, No. 2, 301-308 (1993) · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309