| Title:
|
Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces (English) |
| Author:
|
Saddeek, A. M. |
| Author:
|
Ahmed, Sayed A. |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
44 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
285-293 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The weak convergence of the iterative generated by $J(u_{n+1}-u_{n})= \tau (Fu_{n}-Ju_{n})$, $n \ge 0$, $\big (0< \tau =\min \big \lbrace 1,\frac{1}{\lambda }\big \rbrace \big )$ to a coincidence point of the mappings $F,J\colon V \rightarrow V^{\star }$ is investigated, where $V$ is a real reflexive Banach space and $V^{\star }$ its dual (assuming that $V^{\star }$ is strictly convex). The basic assumptions are that $J$ is the duality mapping, $J-F$ is demiclosed at $0$, coercive, potential and bounded and that there exists a non-negative real valued function $r(u,\eta )$ such that \[ \sup _{u,\eta \in V} \lbrace r(u,\eta )\rbrace =\lambda < \infty \] \[ r(u,\eta )\Vert J(u- \eta ) \Vert _{V^{\star }}\ge \Vert (J -F)(u)-(J-F)(\eta ) \Vert _{V^{\star }}\,, \quad \forall ~ u,\eta \in V\,. \] Furthermore, the case when $V$ is a Hilbert space is given. An application of our results to filtration problems with limit gradient in a domain with semipermeable boundary is also provided. (English) |
| Keyword:
|
iteration |
| Keyword:
|
coincidence point |
| Keyword:
|
demiclosed mappings |
| Keyword:
|
pseudo-monotone mappings |
| Keyword:
|
bounded Lipschitz continuous coercive mappings |
| Keyword:
|
filtration problems |
| MSC:
|
47H10 |
| MSC:
|
54H25 |
| idZBL:
|
Zbl 1212.47088 |
| idMR:
|
MR2493425 |
| . |
| Date available:
|
2009-01-29T09:15:26Z |
| Last updated:
|
2013-09-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119768 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Gajewski, H., Gröger, K., Zacharias, K.: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen.Akademie Verlag Berlin, 1974. MR 0636412 |
| Reference:
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| Reference:
|
[5] Istratescu, V.I.: Fixed Point Theory.Reidel, Dordrecht, 1981. Zbl 0465.47035, MR 0620639 |
| Reference:
|
[6] Karchevskii, M. M., Badriev, I. B.: Nonlinear problems of filtration theory with dis continuous monotone operators.Chislennye Metody Mekh. Sploshnoi Sredy 10 (5) (1979), 63–78, in Russian. MR 0628672 |
| Reference:
|
[7] Lions, J. L.: Quelques Methods de Resolution des Problemes aux Limites Nonlineaires.Dunod and Gauthier-Villars, 1969. MR 0259693 |
| Reference:
|
[8] Lyashko, A. D., Karchevskii, M. M.: On the solution of some nonlinear problems of filtration theory.Izv. Vyssh. Uchebn. Zaved., Matematika 6 (1975), 73–81, in Russian. |
| Reference:
|
[9] Mann, W. R.: Mean value methods in iteration.Proc. Amer. Math. Soc. 4 (1953), 506–510. Zbl 0050.11603, MR 0054846, 10.1090/S0002-9939-1953-0054846-3 |
| Reference:
|
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| Reference:
|
[11] Zeidler, E.: Nonlinear Functional Analysis and Its Applications.Nonlinear Monotone Operators, vol. II(B), Springer Verlag, Berlin, 1990. Zbl 0684.47029 |
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