# Weak and Strong Convergence of Implicit and Explicit Algorithms for Total Asymptotically Nonexpansive Mappings

### Abstract

In this paper, we prove weak and strong convergence of implicit and explicit iterative algorithms for approximation of common fixed point of a finite family of total asymptotically nonexpansive mappings. Our recursion formulas seem more efficient than those recently announced by several authors for the same problem. Our theorems improve, generalize and extend several recently announced results.

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### References

Ya. I. Alber, Metric and generalized operators in Banach spaces, properties, and applications in theory and applications of the nonlinear operator of monotone and accretive type, Dekker, New York, 1996, pp. 15-50.

Ya. I. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings, Fixed point theory and applications, 2006 (2006), Article ID 10673.

Y. Alber, R. Espinola, and P. Lorenzo, Strongly Convergent Approximations to fixed points of total asymptotically nonexpansive mappings, Acta Mathematica Sinica, English Series, vol. 24 no. 6 (2008) 1005-1022.

H. H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl., 202 (1996), 150-159.

F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Zeitschr. 100 (1967) 201-225.

F. E. Browder, Fixed point theorems for nonlinear semicontractive mappings in Banach spaces, Archive for Rat. Mech. and Anal. 21 (1966) 259-269.

F. E. Browder and D. G. De Figueirado, J-monotone nonlinear operators in Banach spaces, Konk. Nederl. Akadi. Wetesch. 69 (1966) 412-420.

S. S. Chang, K. K. Tan, H. W. Joseph Lee, and C. K. Chan, On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings, J. Math. Anal. Appl., 313 (2006) 273-283.

S. S. Chang, H. W. Joseph Lee, and C. K. Chan, On Reich’s strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Nonlinear Analysis, TMA, an article in press, available online 2 August 2006.

C. E. Chidume; Geometric properties of Banach spaces and nonlinear iterations, Springer Verlag Series: Lecture Notes in Mathematics Vol. 1965 (2009), XVII, 326p, ISBN 978-1- 84882-189-7.

C. E. Chidume and E. U. Ofoedu, Approximation of common fixed points for finite families of total asymptotically nonexpansive mappings, J. Math. Anal. Appl. 333 (2007) 128-141.

C. E. Chidume and E. U. Ofoedu, A new iteration process for an approximation of total asymptotically nonexpansive mappings, Int. J. Math. Math. Sci., Volume 2009, Article ID615107, 17 pages, DOI:10.1155/2009/615107.

C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorem for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003), no.2, 364-37.

C. E. Chidume, Jinlu Li and A. Udomene; Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 133 (2005), 473- 480.

C. E. Chidume, H. Zegeye, and N. Shahzad, Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings, Fixed point theory and Application, (2005) 1-9.

C. E. Chidume, Bashir Ali; Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., in the press, available on line 27 April 2006.

Y. J. Cho, G. T. Guo, and H. Y. Zhou, Approximating fixed points of asymptotically quasi nonexpansive mappings by the iterative sequences with errors, Antalya, Turkey-Dynamical systems and applications, proceedings, (5-10 July 2004) 262-272.

T. Figiel, On the moduli of convexity and smoothness, Studia Math., 56 (1976), 121-155.

K. Goebel and W. A. Kirk; A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.

J. P. Gossez and E. Lami Dozo, Some geometric properties related to fixed point theory for nonexpansive mappings, Pacific J. Math. 40 (1972), 565-573.

Y. Hao, Convergence theorems for total asymptotically nonexpansive mappings, An. St. Univ. Ovidus Constanta, vol. 18(1) (2010), 163-180.

J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 302 (2005) 509-520.

J. S. Jung, Y. J. Cho and R. P. Agarwal, Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach spaces, Fixed Point Theory, and Application 2 (2005) 125-135.

S. H. Khan and H. Fukharu-ud-din, Weak and strong convergence of a scheme with error for two nonexpansive mappings, Nonlinear Anal. 61 (2005), 1295-1301.

L. Yang and X. Xie, Weak and strong convergence theorems of three-step iteration process with errors for nonself asymptotically nonexpansive mappings, Mathematical Computer Modelling (2010), DOI:10.1016/j.mcm.2010.05.006.

K. Nammanee, M. A. Noor and S. Suantai, Convergence criteria of modified Noor iterations with errors for asymptotically nonexpansive mappings, J. Math. Anal. Appl. 314 (2006) no.1, 320-334.

J. G. O’Hara, P. Pillay and H. K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Analysis, 54 (2003) 1417-1426.

E. U. Ofoedu, Strong convergence theorem for uniformly L−Lipschitzian asymptotically pseudo-contractive mappings in real Banach Spaces, J. Math. Anal. Appl., 321 (2) (2006), 722-728.

E. U. Ofoedu and L. O. Madu, Iterative Procedures for a finite family of total asymptotically nonexpansive mappings, J. Nig. Math. Soc., 33 (2014), 93-112.

Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the Australian Mathematical Society 73 (1967), 591-597.

S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980)287-292.

D. R. Sahu, Fixed points of semicontinuous nearly Lipshitzian mappings in Banach spaces, Comment. Math. Univ. Carolinae 46, 4 (2005) 653-666.

N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed point Theory, and applications 2006

(2006), Article ID 18909.

T. Shimizu and W. Takahashi; Strong convergence theorems for asymptotically nonexpansive mappings, Nonlinear Analysis, 26 (1996), 265-272.

N. Shioji and W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces Proc. Amer. Math. Soc., 125 (1997), 3641-3645.

N. Shioji and W. Takahashi; A strong convergence theorem for asymptotically nonexpansive mappings in Banach spaces, Arch. Math. 72 (1999), 354-359.

N. Shioji and W. Takahashi; A strong convergence of averaged approximants for asymptotically nonexpansive mappings in Banach spaces, J. Approx. Theory 97 (1999), 53-64.

S. Suantai, Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 311 (2005) no.2, 506-517.

Z. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003) no.1, 351-358.

T. Suzuki, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl., 305 (2005), 227-239.

K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993) no.2, 301-308.

B. Xu and M. A. Noor, Fixed-points iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002), 444-453.

B. L. Xu and M. A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002) no.2, 444-453.

H. Zhou, L. Wei, and Y. J. Cho, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput., 173 (2006), 196-212.

Y. Zhou and S. S. Chang, Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Num. Func. Anal. Opt. vol. 23 no.s 7 and 8 (2002) 911-921.

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