IJPAM: Volume 11, No. 1 (2004)

GENERAL TWO-STEP MODELS FOR PROJECTION
METHODS AND THEIR APPLICATIONS

Ram U. Verma
International Publications
5066 Jamieson Drive, Suite B-9
Toledo, Ohio 43613, USA
e-mail: verma99@msn.com


Abstract.This paper concerns the general two-step model for projection methods and its applications to the approximation solvability of a system of nonlinear variational problems in general Hilbert spaces. First, a general model for two-step projection methods is introduced and second, it is applied to the approximation solvability of a system of nonlinear variational inequalities in a Hilbert space setting. Let $H$ be a real Hilbert space and $K$ be a nonempty closed convex subset of $H$. For arbitrarily chosen initial points $x^{0}$, $y^{0} \in K$, update sequences {$x^{k}$} and {$y^{k}$} iteratively such that

\begin{displaymath} x^{k + 1 }= (1 - a^{k})x^{k }+ a^{k}P_{K}[y^{k} - \rho T(y^{k})]\,,\ \ \text{for}\ \ \rho > 0\,, \end{displaymath}


\begin{displaymath} y^{k }= (1 - b^{k})x^{k }+ b^{k}P_{K}[x^{k } - \eta T(x^{k})] \,,\ \ \text{for}\ \ \eta > 0, \end{displaymath}

where $T: K \longrightarrow H$ is a nonlinear mapping on $K$, $P_{K }$ is the projection of $H$ onto $K$, and sequences {$a^{k}$}and {$b^{k}$} satisfy $0 \le a^{k}$, $b^{k}\le 1$.

Received: September 23, 2003

AMS Subject Classification: 49J40, 65B05

Key Words and Phrases: general two-step model, system of strongly monotone nonlinear variational inequalities, projection methods, convergence of two-step projection methods

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 11
Issue: 1