| Title:
|
Some fixed point theorems in logarithmic convex structures (English) |
| Author:
|
Moazzen, Alireza |
| Author:
|
Cho, Yoel-Je |
| Author:
|
Park, Choonkil |
| Author:
|
Eshaghi Gordji, Madjid |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
142 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
1-7 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we introduce the concept of a logarithmic convex structure. Let $X$ be a set and $D\colon X\times X\rightarrow [1,\infty )$ a function satisfying the following conditions: \item {(i)} For all $x,y\in X$, $ D(x,y)\geq 1$ and $D(x,y)=1$ if and only if $x=y$. \item {(ii)} For all $x,y\in X$, $D(x,y)=D(y,x)$. \item {(iii)} For all $ x,y,z\in X$, $D(x,y)\leq D(x,z)D(z,y)$. \item {(iv)} For all $x,y,z\in X$, $z\neq x,y$ and $\lambda \in (0,1)$, \begin {gather} D(z,W(x,y,\lambda ))\leq D^\lambda (x,z)D^{1-\lambda }(y,z),\nonumber \\ D(x,y)= D(x,W(x,y,\lambda ))D(y,W(x,y,\lambda )),\nonumber \end {gather} where $W\colon X\times X\times [0,1]\rightarrow X$ is a continuous mapping. We name this the logarithmic convex structure. In this work we prove some fixed point theorems in the logarithmic convex structure. (English) |
| Keyword:
|
fixed point |
| Keyword:
|
logarithmic convex structure |
| Keyword:
|
convex metric space |
| MSC:
|
47H09 |
| MSC:
|
47H10 |
| MSC:
|
54H25 |
| idZBL:
|
Zbl 06738565 |
| idMR:
|
MR3619982 |
| DOI:
|
10.21136/MB.2017.0074-14 |
| . |
| Date available:
|
2017-02-21T17:19:04Z |
| Last updated:
|
2020-07-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146002 |
| . |
| Reference:
|
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| Reference:
|
[2] 'Cirić, L. B.: On some discontinuous fixed point mappings in convex metric spaces.Czech. Math. J. 43 (1993), 319-326. Zbl 0814.47065, MR 1211753 |
| Reference:
|
[3] Guay, M. D., Singh, K. L., Whitfieled, J. H. M.: Fixed point theorems for nonexpansive mappings in convex metric spaces.Nonlinear Analysis and Applications. Proc. Int. Conf. at Memorial University of Newfoundland, 1981 S. P. Singh at al. Lect. Notes Pure Appl. Math. 80, Marcel Dekker, New York (1982), 179-189. Zbl 0501.54030, MR 0689554 |
| Reference:
|
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| Reference:
|
[5] Shimizu, T., Takahashi, W.: Fixed point theorems in certain convex metric spaces.Math. Jap. 37 (1992), 855-859. Zbl 0764.47030, MR 1186552 |
| Reference:
|
[6] Takahashi, W.: A convexity in metric spaces and nonexpansive mapping I.Kōdai Math. Semin. Rep. 22 (1970), 142-149. Zbl 0268.54048, MR 0267565, 10.2996/kmj/1138846111 |
| Reference:
|
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