Iterative Solution of Nonlinear Equations of the Pseudo-monotone Type in Banach Spaces

The weak convergence of the iterative generated by J(un+1 − un) = τ(Fun − Jun), n ≥ 0, ( 0 < τ = min { 1, 1 λ }) to a coincidence point of the mappings F, J : V → V ? is investigated, where V is a real reflexive Banach space and V ? its dual (assuming that V ? 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, η) such that sup u,η∈V {r(u, η)} = λ <∞ r(u, η)‖J(u− η)‖V ? ≥ ‖(J − F )(u)− (J − F )(η)‖V ? , ∀ u, η ∈ 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.