Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces

Saddeek, A. M., and Ahmed, Sayed A.. "Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces." Archivum Mathematicum 044.4 (2008): 285-293. <http://eudml.org/doc/250462>.

@article{Saddeek2008,
abstract = {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.},
author = {Saddeek, A. M., Ahmed, Sayed A.},
journal = {Archivum Mathematicum},
keywords = {iteration; coincidence point; demiclosed mappings; pseudo-monotone mappings; bounded Lipschitz continuous coercive mappings; filtration problems; iteration; coincidence point; demiclosed mapping; pseudo-monotone mapping; bounded Lipschitz continuous coercive mapping; filtration problem},
language = {eng},
number = {4},
pages = {285-293},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces},
url = {http://eudml.org/doc/250462},
volume = {044},
year = {2008},
}

TY - JOUR
AU - Saddeek, A. M.
AU - Ahmed, Sayed A.
TI - Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces
JO - Archivum Mathematicum
PY - 2008
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 044
IS - 4
SP - 285
EP - 293
AB - 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.
LA - eng
KW - iteration; coincidence point; demiclosed mappings; pseudo-monotone mappings; bounded Lipschitz continuous coercive mappings; filtration problems; iteration; coincidence point; demiclosed mapping; pseudo-monotone mapping; bounded Lipschitz continuous coercive mapping; filtration problem
UR - http://eudml.org/doc/250462
ER -