Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings

Zeqing Liu; Ravi P. Agarwal; Chi Feng; Shin Min Kang

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica (2005)

  • Volume: 44, Issue: 1, page 83-96
  • ISSN: 0231-9721

Abstract

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The purpose of this paper is to establish some weak and strong convergence theorems of modified three-step iteration methods with errors with respect to a pair of nonexpansive and asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize, improve and unify a few results due to Chang [1], Liu and Kang [5], Osilike and Aniagbosor [7], Rhoades [8] and Schu [9], [10] and others. An example is included to demonstrate that our results are sharp.

How to cite

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Liu, Zeqing, et al. "Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings." Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica 44.1 (2005): 83-96. <http://eudml.org/doc/32447>.

@article{Liu2005,
abstract = {The purpose of this paper is to establish some weak and strong convergence theorems of modified three-step iteration methods with errors with respect to a pair of nonexpansive and asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize, improve and unify a few results due to Chang [1], Liu and Kang [5], Osilike and Aniagbosor [7], Rhoades [8] and Schu [9], [10] and others. An example is included to demonstrate that our results are sharp.},
author = {Liu, Zeqing, Agarwal, Ravi P., Feng, Chi, Kang, Shin Min},
journal = {Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica},
keywords = {nonexpansive mappings; asymptotically nonexpansive mappings; common fixed points; modified three-step iteration methods with errors with respect to a pair of mappings; nonexpansive mapping; asymptotically nonexpansive mapping; fixed point},
language = {eng},
number = {1},
pages = {83-96},
publisher = {Palacký University Olomouc},
title = {Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings},
url = {http://eudml.org/doc/32447},
volume = {44},
year = {2005},
}

TY - JOUR
AU - Liu, Zeqing
AU - Agarwal, Ravi P.
AU - Feng, Chi
AU - Kang, Shin Min
TI - Weak and strong convergence theorems of common fixed points for a pair of nonexpansive and asymptotically nonexpansive mappings
JO - Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica
PY - 2005
PB - Palacký University Olomouc
VL - 44
IS - 1
SP - 83
EP - 96
AB - The purpose of this paper is to establish some weak and strong convergence theorems of modified three-step iteration methods with errors with respect to a pair of nonexpansive and asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize, improve and unify a few results due to Chang [1], Liu and Kang [5], Osilike and Aniagbosor [7], Rhoades [8] and Schu [9], [10] and others. An example is included to demonstrate that our results are sharp.
LA - eng
KW - nonexpansive mappings; asymptotically nonexpansive mappings; common fixed points; modified three-step iteration methods with errors with respect to a pair of mappings; nonexpansive mapping; asymptotically nonexpansive mapping; fixed point
UR - http://eudml.org/doc/32447
ER -

References

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  1. Chang S. S., On the approximation problem of fixed points for asymptotically nonexpansive mappings, Indian J. Pure Appl. Math. 32 (2001), 1297–1307. Zbl1034.47039
  2. Chang S. S., Some problems and results in the study of nonlinear analysis, Nonlinear Anal. TMA 30 (1997), 4197–4208. (1997) Zbl0901.47036
  3. Goebel K., Kirk W. A., A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174. (1972) Zbl0256.47045
  4. Gornicki J., Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces, Comment. Math. Univ. Carolina 30 (1989), 249–252. (1989) Zbl0686.47045
  5. Liu Z., Kang S. M., Weak and strong convergence for fixed points of asymptotically nonexpansive mappings, Acta. Math. Sinica 20 (2004), 1009–1018. 
  6. Opial Z., Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597. (1967) Zbl0179.19902
  7. Osilike M. O., Aniagbosor S. C., Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. Comput. Modelling 32 (2000), 1181–1191. Zbl0971.47038
  8. Rhoades B. E., Fixed point iteration for certain nonlinear mappings, J. Math. Anal. Appl. 183 (1994), 118–120. (1994) 
  9. Schu J., Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 407–413. (1991) Zbl0734.47036
  10. Schu J., Weak and strong convergence of fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159. (1991) 
  11. Xu H. K., Inequalities in Banach spaces with applications, Nonlinear Anal. TMA 16 (1991), 1127–1138. (1991) Zbl0757.46033

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