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

## A.M. Saddeek and Sayed A. Ahmed

Address: Department of Mathematics, Faculty of Science Assiut University, Assiut, Egypt

E-mail:

a_m_saddeek@yahoo.com

s_a_ahmed2003@yahoo.com

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
\[ \sup _{u,\eta \in V} \lbrace r(u,\eta )\rbrace =\lambda < \infty \\[3pt] r(u,\eta )\Vert J(u- \eta ) \Vert _{V^{\star }}\ge \Vert (J -F)(u)-(J-F)(\eta ) \Vert _{V^{\star }}\,, \quad \forall ~ u,\eta \in V\,. \]
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.

AMSclassification: primary 47H10; secondary 54H25.

Keywords: iteration, coincidence point, demiclosed mappings, pseudo-monotone mappings, bounded Lipschitz continuous coercive mappings, filtration problems.