# Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

- Ram U. Verma
^{1}Email author

**2006**:90295

**DOI: **10.1155/JIA/2006/90295

© Verma 2006

**Received: **30 April 2004

**Accepted: **29 August 2004

**Published: **18 January 2006

## Abstract

Let
be a nonlinear mapping from a nonempty closed invex subset
of an infinite-dimensional Hilbert space
into
. Let
be proper, invex, and lower semicontinuous on
and let
be continuously Fréchet-differentiable on
with
, the gradient of
,
-*strongly* monotone, and
-*Lipschitz* continuous on
. Suppose that there exist an
, and numbers
,
,
such that for all
and for all
, the set
defined by
is nonempty, where
and
is
-*Lipschitz* continuous with the following assumptions. (i)
, and
. (ii) For each fixed
, map
is sequentially continuous from the weak topology to the weak topology. If, in addition,
is continuous from
equipped with weak topology to
equipped with strong topology, then the sequence
generated by the general auxiliary problem principle converges to a solution
of the variational inequality problem (VIP):
for all
.

## Authors’ Affiliations

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