# MODIFIED ISHIKAWA ITERATIVE SEQUENCES WITH ERRORS FOR ASYMPTOTICALLY SET-VALUED PSEUCOCONTRACTIVE MAPPINGS IN BANACH SPACES

• Kim, Jong-Kyu (DEPARTMENT OF MATHEMATICS, EDUCATION, KYUNGNAM UNIVERSITY) ;
• Nam, Young-Man (DEPARTMENT OF MATHEMATICS, EDUCATION, KYUNGNAM UNIVERSITY)
• Published : 2006.11.30

#### Abstract

In this paper, some new convergence theorems of the modified Ishikawa and Mann iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in uniformly smooth Banach spaces are given.

#### References

1. R. P. Agarwal, N. J. Huang, and Y. J. Cho, Stability of iterative processes with errors for nonlinear equations of $\phi$-strongly accretive type operators, Numer. Funct. Anal. Optim. 22 (2001), no. 5-6, 471-485 https://doi.org/10.1081/NFA-100105303
2. E. Asplund, Positivity of duality mappings, Bull. Amer. Math. Soc. 73 (1967), 200-203 https://doi.org/10.1090/S0002-9904-1967-11678-1
3. F. E. Browder, Nonexpansive nonlinear operators in Banach spaces, Proc. Natl. Acad. Sci. U. S. A. 54 (1965), 1041-1044
4. S. S. Chang, Convergence of Iterative methods for accretive and pseudo-contr-active type mappings in Banach spaces, Nonlinear Funct. Anal. Appl. 4 (1999), 1-23
5. S. S. Chang, Some results for asymptotically pseudo-contractive mappings and asy-mptotically nonexpansive mappings, Proc. Amer. Math. Soc. 129 (2001), no. 3, 845-853
6. S. S. Chang, On the approximatiing problems of fixed points for asymptotically non-expansive mappings, Indian J. Pure Appl. Math. 32 (2001), no. 9, 1297-1307
7. S. S. Chang, J. K. Kim, and Y. J. Cho, Approximations of solutions for set-valued $\phi$-strongly accretive equations, Dynam. Systems Appl. 14 (2005), no. 3-4, 515-524
8. Y. P. Fang, J. K. Kim, and N. J. Huang, Stable iterative procedures with errors for strong pseudocontractions and nonlinear equations of accretive operators without Lipschitz assumptions, Nonlinear Funct. Anal. Appl. 7 (2002), no. 4, 497-507
9. M. K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasi-nonexp-ansive mappings, J. Math. Anal. Appl. 207 (1997), no. 1, 96-103 https://doi.org/10.1006/jmaa.1997.5268
10. K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpan-sive mappings, Proc. Amer. Math. Soc. 35 (1972), no. 1, 171-174
11. N. J. Huang and M. R. Bai, A perturbed iterative procedure for multivalued pseudo-contractive mappings and multivalued accretive mappings in Banach Spaces, Comput. Math. Appl. 37 (1999), no. 6, 7-15 https://doi.org/10.1016/S0898-1221(99)00072-3
12. J. K. Kim, Convergence of Ishikawa iterative sequences for accretive Lipschitz-ian mappings in Banach spaces, Taiwan Jour. Math. 10 (2006), no. 2, 553-561 https://doi.org/10.11650/twjm/1500403843
13. J. K. Kim, S. M. Jang, and Z. Liu, Convergence theorems and stability problems of Ishikawa iterative sequences for nonlinear operator equations of the accretive and strong accretive operators, Comm. Appl. Nonlinear Anal. 10 (2003), no. 3, 85-98
14. J. K. Kim, Z. Liu, and S. M. Kang, Almost stability of Ishikawa iterative schemes with errors for $\phi$-strongly quasi-accretive and $\phi$-hemicontractive operators, Commun. Korean Math. Soc. 19 (2004), no. 2, 267-281 https://doi.org/10.4134/CKMS.2004.19.2.267
15. J. K. Kim, Z. Liu, Y. M. Nam, and S. A. Chun, Strong convergence theorems and stability problems of Mann and Ishikawa iterative sequences for strictly hemi-contractive mappings, J. Nonlinear and Convex Anal. 5 (2004), no. 2, 285-294
16. W. A. Kirk, A fixed point theorem for mappings which do not increase distance, Amer. Math. Monthly 72 (1965), 1004-1006 https://doi.org/10.2307/2313345
17. Q. H. Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal. 26 (1996), no. 11, 1835-1842 https://doi.org/10.1016/0362-546X(94)00351-H
18. W. V. Petryshyn, A characterization of strict convexity of Banach spaces and other uses of duality mappings, J. Funct. Anal. 6 (1970), 282-291 https://doi.org/10.1016/0022-1236(70)90061-3
19. B. E. Rhoades, Comments on two fixed point iterative methods, J. Math. Anal. Appl. 56 (1976), no. 3, 741-750 https://doi.org/10.1016/0022-247X(76)90038-X
20. J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 407-413 https://doi.org/10.1016/0022-247X(91)90245-U
21. H. K. Xu, Inequalities in Banach spaces with applications, Nonlinear Anal. 16 (1991), no. 12, 1127-1138 https://doi.org/10.1016/0362-546X(91)90200-K
22. H. K. Xu, Existence and convergence for fixed points of mapping of asymptotically nonexpansive type, Nonlinear Anal. 16 (1991), no. 12, 1139-1146 https://doi.org/10.1016/0362-546X(91)90201-B
23. Y. G. Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, J. Math. Anal. Appl. 224 (1998), 91-101 https://doi.org/10.1006/jmaa.1998.5987

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2. Shrinking Projection Method of Fixed Point Problems for Asymptotically Pseudocontractive Mapping in the Intermediate Sense and Mixed Equilibrium Problems in Hilbert Spaces vol.2012, 2012, https://doi.org/10.1155/2012/187421