| Title:
|
On Diviccaro, Fisher and Sessa open questions (English) |
| Author:
|
Ćirić, Ljubomir B. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
29 |
| Issue:
|
2 |
| Year:
|
1993 |
| Pages:
|
145-152 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $K$ be a closed convex subset of a complete convex metric space $X$ and $T, I: K \rightarrow K$ two compatible mappings satisfying following contraction definition: $Tx, Ty)\le (Ix, Iy)+(1-a)\max \ \lbrace Ix.Tx),\ Iy, Ty)\rbrace $ for all $x,y$ in $K$, where $0<a<1/2^{p-1}$ and $p\ge 1$. If $I$ is continuous and $I(K)$ contains $[T(K)]$ , then $T$ and $I$ have a unique common fixed point in $K$ and at this point $T$ is continuous. This result gives affirmative answers to open questions set forth by Diviccaro, Fisher and Sessa in connection with necessarity of hypotheses of linearity and non-expansivity of $I$ in their Theorem [3] and is a generalisation of that Theorem. Also this result generalizes theorems of Delbosco, Ferrero and Rossati [2], Fisher and Sessa [4], Gregus [5], G. Jungck [7] and Mukherjee and Verma [8]. Two examples are presented, one of which shows the generality of this result. (English) |
| Keyword:
|
convex metric space |
| Keyword:
|
Cauchy sequence |
| Keyword:
|
fixed point |
| MSC:
|
47H10 |
| MSC:
|
54H25 |
| idZBL:
|
Zbl 0810.47051 |
| idMR:
|
MR1263115 |
| . |
| Date available:
|
2008-06-06T21:24:25Z |
| Last updated:
|
2012-05-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/107476 |
| . |
| Reference:
|
[1] Ćirić, Lj.B.: On a common fixed point theorem of a Greguš type.Publ. Inst. Math. 49(63) (1991), Beograd, 174-178. MR 1127395 |
| Reference:
|
[2] Delbosco, D., Ferrero, O., Rossati, F.: Teoreme di punto fisso per applicazioni negli spazi di Banach.Boll. Un. Mat. Ital. (6) 2-A (1983), 297-303. MR 0724481 |
| Reference:
|
[3] Diviccaro, M. L., Fisher, B., Sessa, S.: A common fixed point theorem of Greguš type.Publ. Math. Debrecen 34 (1987), No. 1-2. MR 0901008 |
| Reference:
|
[4] Fisher, B., Sessa, S.: On a fixed point theorem of Greguš.Internat. J. Math. Math. 9 (1986), No. 1, 23-28. MR 0837098 |
| Reference:
|
[5] Greguš, M.: A fixed point theorem in Banach space.Boll. Un. Mat. Ital. (5) 7-A (1980), 193-198. MR 0562137 |
| Reference:
|
[6] Jungck, G.: Compatible mappings and common fixed points.Internat. J. Math. Math. Sci. 9 (1986), 771-779. MR 0870534 |
| Reference:
|
[7] Jungck, G.: On a fixed point theorem of Fisher and Sessa.Internat. J. Math. Math. Sci 13 (1988), 497-500. MR 1068012 |
| Reference:
|
[8] Mukherjee, R. N., Verma, V.: A note on a fixed point theorem of Greguš.Math. Japon. 33 (1988), 745-749. MR 0972387 |
| Reference:
|
[9] Sessa, S.: On a week commutativity condition in fixed point considerations.Publ. Inst. Math. (Beograd) (N.S.) 32(46) (1982), 149-153. MR 0710984 |
| Reference:
|
[10] Takahashi, W.: A convexity in metric space and nonexpansive mappings $I$.Kodai Math. Sem. Rep. 22 (1970), 142-149. Zbl 0268.54048, MR 0267565 |
| . |
