IJPAM: Volume 1, No. 1 (2002)

APPROXIMATION SOLVABILITY OF
NONLINEAR VARIATIONAL INEQUALITIES
BASED ON GENERAL AUXILIARY
PROBLEM PRINCIPLE

Ram U. VermaUniversity of Toledo, Dept. of Mathematics, Toledo, Ohio 43606, USA
International Publications, USA
12046 Coed Drive, Suite A-29
Orlando, Florida 32826, USA
e-mail: verma99@msn.com


Abstract.First a general class of auxiliary problem principle is introduced and then it is applied to approximation-solvability of the following class of nonlinear variational inequality problems (NVIP):

Find an element $x^* \in K$ such that

\begin{displaymath}
\langle T(x^*), x - x^* \rangle + f(x) - f(x^*) \geq 0\,,\quad \text{for all}\ x \in K,
\end{displaymath}

where $T: K \longrightarrow E^*$ is a mapping from a nonempty closed convex subset $K$ of a reflexive Banach space $E$ into its dual $E^*$, and $f: K \longrightarrow R$ is a continuous convex functional on $K$. This general class of auxiliary problem principle is described as follows: for a given iterate $x^k \in K$ and for a constant $\rho > 0$, determine $x^{k+1}$ such that
\begin{multline*}
\langle\rho T(x^{k}) + h^\prime (x^{k+1}) - h^\prime (x^{k}),...
... 0\,,\\
\text{for all}\ x \in K\,,\ \text{and for}\ k \geq 0,
\end{multline*}
where $h: K \longrightarrow R$ is $m$-times continuously Frechet-differentiable $(m \geq 2)$ on $K$.

Received: July 22, 2001

AMS Subject Classification: 49J40

Key Words and Phrases: general auxiliary problem principle, auxiliary variational inequality problem, approximation-solvability, approximate solutions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2002
Volume: 1
Issue: 1