Geodesic
Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces
Archivum mathematicum, Tome 44 (2008) no. 4, pp. 285-293
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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.
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.
Classification : 47H10, 54H25
Keywords: iteration; coincidence point; demiclosed mappings; pseudo-monotone mappings; bounded Lipschitz continuous coercive mappings; filtration problems 
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Saddeek, A. M.; Ahmed, Sayed A. Iterative solution of nonlinear equations of the pseudo-monotone type in Banach spaces. Archivum mathematicum, Tome 44 (2008) no. 4, pp. 285-293. http://geodesic.mathdoc.fr/item/ARM_2008_44_4_a3/

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