| Title:
|
Convergence of Ishikawa iterates for a multi-valued mapping with a fixed point (English) |
| Author:
|
Sastry, K. P. R. |
| Author:
|
Babu, G. V. R. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
55 |
| Issue:
|
4 |
| Year:
|
2005 |
| Pages:
|
817-826 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Existence of fixed points of multivalued mappings that satisfy a certain contractive condition was proved by N. Mizoguchi and W. Takahashi. An alternative proof of this theorem was given by Peter Z. Daffer and H. Kaneko. In the present paper, we give a simple proof of that theorem. Also, we define Mann and Ishikawa iterates for a multivalued map $T$ with a fixed point $p$ and prove that these iterates converge to a fixed point $q$ of $T$ under certain conditions. This fixed point $q$ may be different from $p$. To illustrate this phenomenon, an example is given. (English) |
| Keyword:
|
multi-valued map |
| Keyword:
|
Mann iterates |
| Keyword:
|
Ishikawa iterates |
| Keyword:
|
fixed points |
| MSC:
|
47H04 |
| MSC:
|
47H10 |
| MSC:
|
47J25 |
| MSC:
|
54C60 |
| MSC:
|
54H25 |
| idZBL:
|
Zbl 1081.47069 |
| idMR:
|
MR2184365 |
| . |
| Date available:
|
2009-09-24T11:28:04Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128026 |
| . |
| Reference:
|
[1] P. Z. Daffer and H. Kaneko: Fixed points of generalized contractive multi-valued mappings.J. Math. Anal. Appl. 192 (1995), 655–666. MR 1332232, 10.1006/jmaa.1995.1194 |
| Reference:
|
[2] R. L. Franks and R. P. Marzec: A theorem on mean value iterations.Proc. Amer. Math. Soc. 30 (1971), 324–326. MR 0280656, 10.1090/S0002-9939-1971-0280656-9 |
| Reference:
|
[3] S. Ishikawa: Fixed points by a new iteration method.Proc. Amer. Math. Soc. 44 (1974), 147–150. Zbl 0286.47036, MR 0336469, 10.1090/S0002-9939-1974-0336469-5 |
| Reference:
|
[4] A. K. Kalinde and B. E. Rhoades: Fixed point Ishikawa iterations.J. Math. Anal. Appl. 170 (1992), 600–606. MR 1188575, 10.1016/0022-247X(92)90040-K |
| Reference:
|
[5] H. Kaneko: Generalized contractive multi-valued mappings and their fixed points.Math. Japon. 33 (1988), 57–64. MR 0935499 |
| Reference:
|
[6] W. R. Mann: Mean value methods in iterations.Proc. Amer. Math. Soc. 4 (1953), 506–510. MR 0054846, 10.1090/S0002-9939-1953-0054846-3 |
| Reference:
|
[7] N. Mizoguchi and W. Takahashi: Fixed point theorems for multi-valued mappings on complete metric spaces.J. Math. Anal. Appl. 141 (1989), 177–188. MR 1004592, 10.1016/0022-247X(89)90214-X |
| Reference:
|
[8] Liu Qihou: On Naimpally and Singh’s open questions.J. Math. Anal. Appl. 124 (1987), 157–164. Zbl 0625.47044, MR 0883519, 10.1016/0022-247X(87)90031-X |
| Reference:
|
[9] Liu Qihou: A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings.J. Math. Anal. Appl. 146 (1990), 301–305. Zbl 0721.47042, MR 1043101, 10.1016/0022-247X(90)90303-W |
| Reference:
|
[10] B. E. Rhoades: Comments on two fixed point iteration methods.J. Math. Anal. Appl. 56 (1976), 741–750. Zbl 0353.47029, MR 0430880, 10.1016/0022-247X(76)90038-X |
| . |
