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.
@article{119768,
author = {A. M. Saddeek and Sayed A. Ahmed},
title = {Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces},
journal = {Archivum Mathematicum},
volume = {044},
year = {2008},
pages = {285-293},
zbl = {1212.47088},
mrnumber = {2493425},
language = {en},
url = {http://dml.mathdoc.fr/item/119768}
}
Saddeek, A. M.; Ahmed, Sayed A. Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces. Archivum Mathematicum, Tome 044 (2008) pp. 285-293. http://gdmltest.u-ga.fr/item/119768/
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