| Title:
|
Random fixed point theorems for a certain class of mappings in Banach spaces (English) |
| Author:
|
Jung, Jong Soo |
| Author:
|
Cho, Yeol Je |
| Author:
|
Kang, Shin Min |
| Author:
|
Lee, Byung Soo |
| Author:
|
Thakur, Balwant Singh |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
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1572-9141 (online) |
| Volume:
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50 |
| Issue:
|
2 |
| Year:
|
2000 |
| Pages:
|
379-396 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $(\Omega,\Sigma)$ be a measurable space and $C$ a nonempty bounded closed convex separable subset of $p$-uniformly convex Banach space $E$ for some $p > 1$. We prove random fixed point theorems for a class of mappings $T\: \Omega \times C \rightarrow C$ satisfying: for each $x, y \in C$, $\omega \in \Omega $ and integer $n \ge 1$, \[\Vert T^n(\omega , x) - T^n(\omega , y) \Vert \le a(\omega )\cdot \Vert x - y \Vert + b(\omega )\lbrace \Vert x - T^n(\omega ,x) \Vert + \Vert y - T^n(\omega ,y) \Vert \rbrace + c(\omega )\lbrace \Vert x - T^n(\omega ,y) \Vert + \Vert y - T^n(\omega ,x) \Vert \rbrace , \] where $a,b,c\: \Omega \rightarrow [0, \infty )$ are functions satisfying certain conditions and $T^n(\omega ,x)$ is the value at $x$ of the $n$-th iterate of the mapping $T(\omega ,\cdot )$. Further we establish for these mappings some random fixed point theorems in a Hilbert space, in $L^p$ spaces, in Hardy spaces $H^p$ and in Sobolev spaces $H^{k,p} $ for $1 < p < \infty $ and $k \ge 0$. As a consequence of our main result, we also extend the results of Xu [43] and randomize the corresponding deterministic ones of Casini and Maluta [5], Goebel and Kirk [13], Tan and Xu [37], and Xu [39, 41]. (English) |
| Keyword:
|
$p$-uniformly convex Banach space |
| Keyword:
|
normal structure |
| Keyword:
|
asymptotic center |
| Keyword:
|
random fixed points |
| Keyword:
|
generalized random uniformly Lipschitzian mapping |
| MSC:
|
47H09 |
| MSC:
|
47H10 |
| MSC:
|
47H40 |
| MSC:
|
60H25 |
| idZBL:
|
Zbl 1048.47043 |
| idMR:
|
MR1761395 |
| . |
| Date available:
|
2009-09-24T10:33:44Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/127577 |
| . |
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