Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings

Kittipong Sitthikul; Satit Saejung

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2009)

  • Volume: 48, Issue: 1, page 139-152
  • ISSN: 0231-9721

Abstract

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In this paper, weak and strong convergence of finite step iteration sequences to a common fixed point for a pair of a finite family of nonexpansive mappings and a finite family of asymptotically nonexpansive mappings in a nonempty closed convex subset of uniformly convex Banach spaces are presented.

How to cite

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Sitthikul, Kittipong, and Saejung, Satit. "Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 48.1 (2009): 139-152. <http://eudml.org/doc/35183>.

@article{Sitthikul2009,
abstract = {In this paper, weak and strong convergence of finite step iteration sequences to a common fixed point for a pair of a finite family of nonexpansive mappings and a finite family of asymptotically nonexpansive mappings in a nonempty closed convex subset of uniformly convex Banach spaces are presented.},
author = {Sitthikul, Kittipong, Saejung, Satit},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {Nonexpansive mapping; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence; weak convergence; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence},
language = {eng},
number = {1},
pages = {139-152},
publisher = {Palacký University Olomouc},
title = {Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings},
url = {http://eudml.org/doc/35183},
volume = {48},
year = {2009},
}

TY - JOUR
AU - Sitthikul, Kittipong
AU - Saejung, Satit
TI - Convergence theorems for a finite family of nonexpansive and asymptotically nonexpansive mappings
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2009
PB - Palacký University Olomouc
VL - 48
IS - 1
SP - 139
EP - 152
AB - In this paper, weak and strong convergence of finite step iteration sequences to a common fixed point for a pair of a finite family of nonexpansive mappings and a finite family of asymptotically nonexpansive mappings in a nonempty closed convex subset of uniformly convex Banach spaces are presented.
LA - eng
KW - Nonexpansive mapping; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence; weak convergence; asymptotically nonexpansive mapping; common fixed point; finite-step iterative sequence
UR - http://eudml.org/doc/35183
ER -

References

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  1. Chidume, C. E., Ali, B., Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 326, 2 (2007), 960–973. (2007) Zbl1112.47053MR2280956
  2. Chidume, C. E., Ali, B., Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 330, 1 (2007), 377–387. (2007) Zbl1141.47039MR2302930
  3. Cho, Y. J., Zhou, H., Guo, G., Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl. 47, 4-5 (2004), 707–717. (2004) Zbl1081.47063MR2051342
  4. Clarkson, J. A., Uniformly convex spaces, Trans. Amer. Math. Soc. 40, 3 (1936), 396–414. (1936) Zbl0015.35604MR1501880
  5. García-Falset, J., Kaczor, W., Kuczumow, T., Reich, S., Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., Ser. A: Theory Methods 43, 3 (2001), 377–401. (2001) MR1796983
  6. Goebel, K., Kirk, W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. (1972) Zbl0256.47045MR0298500
  7. Gornicki, J., Weak and strong convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment. Math. Univ. Carolin 30 (1989), 249–252. (1989) MR1014125
  8. Ishikawa, S., Fixed points by new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147–150. (1974) MR0336469
  9. Liu, Z., Agarwal, R. P., Feng, C., Kang, S. M., Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings, Acta Univ. Palacki. Olomuc., Fac. rer. nat., Math. 44 (2005), 83–96. (2005) Zbl1099.47057MR2218570
  10. Liu, Z., Feng, C., Ume, J. S., Kang, S. M., Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings, Taiwanese J. Math. 11, 1 (2007), 27–42. (2007) Zbl1151.47060MR2304002
  11. Mann, W. R., Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510. (1953) Zbl0050.11603MR0054846
  12. Opial, Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. (1967) Zbl0179.19902MR0211301
  13. Osilike, M. O., Aniagbosor, S. C., Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32, 10 (2000), 1181–1191. (2000) Zbl0971.47038MR1791754
  14. Schu, J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158, 2 (1991), 407–413. (1991) Zbl0734.47036MR1117571
  15. Schu, J., Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43, 1 (1991), 153–159. (1991) Zbl0709.47051MR1086729
  16. Senter, H. F., Dotson, W. G., Jr, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375–380. (1974) Zbl0299.47032MR0346608
  17. Tan, K. K., Xu, H. K., Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122 (1994), 733–739. (1994) Zbl0820.47071MR1203993
  18. Thakur, B. S., Jung, J. S., Weak and strong convergence of multistep iteration for finite family of asymptotically nonexpansive mappings, Fixed Point Theory Appl., Art. ID 31056 (2007), 15 pp. (2007) Zbl1155.47315MR2335967
  19. Xu, B., Noor, M. A., Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 267 (2002), 444–453. (2002) Zbl1011.47039MR1888015
  20. Zhou, Y. Y., Chang, S. S., Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces, Numer. Funct. Anal. and Optimiz. 23 (2002), 911–921. (2002) MR1938840

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