IJPAM: Volume 22, No. 1 (2005)

PARTIALLY RELAXED PSEUDOMONOTONE MAPPINGS
IN APPROXIMATION-SOLVABILITY OF NONLINEAR
VARIATIONAL INEQUALITIES

Ram U. Verma
Department of Theoretical and Applied Mathematics
The University of Akron
Akron, OH 44325, USA
e-mail: verma99@msn.com


Abstract.Let $H$ be a real (finite-dimensional) Hilbert space and $K$ be a nonempty closed convex subset of $H.$ Let $T: K\to H$ be $(\gamma)-partially$ relaxed pseudomonotone, and let $L:K \to H$ be $(\delta, r)- partially$ relaxed pseudomonotone. Then a class of nonlinear variational inequality (NVI) problems is described as: find an element $x^{*}\in K$ such that

\begin{displaymath}\langle \rho (T(x^*) + L(x^*)),x-x^*\rangle \geq 0\,,\ \ \forall\ x\in K \ \text{and for}\ \rho > 0. \end{displaymath}

Let $x^*$ be a solution to the NVI problem and a sequence $\{x^k\}$ be generated by a certain iterative algorithm. Suppose that mappings $T,L:K\to H$ satisfy the following assumptions:
(i)
$T$ is $(\gamma)-partially$ relaxed pseudomonotone.
(ii)
$T$ is $(\beta)-Lipschitz$ continuous.
(iii)
$L$ is $\delta, r)- partially$ relaxed pseudomonotone.
(iv)
$L$ is $(s)-Lipschitz$ continuous.
Then the sequence $\{x^k\}$ converges to $x^*$ for $0<\rho< \frac{1}{2r},$ $0<\rho< \frac{1}{2(\gamma + \delta)}$, and the following estimates hold:
(a)
$\Vert x^{k+1}-x^*\Vert^2 \leq (1-2\rho r)\Vert x^k-x^*\Vert^2 -[1-2\rho(\gamma+\delta)]\Vert x^k-x^{k+1}\Vert^2.$
(b)
$\Vert x^{k+1}-x^*\Vert^2 \leq \Vert x^k-x^*\Vert^2 -[1-2\rho(\gamma+\delta)]\Vert x^k-x^{k+1}\Vert^2.$


Received: January 20, 2005

AMS Subject Classification: 49J40, 65B05

Key Words and Phrases: partially relaxed pseudomonotone mappings, approximation-solvability, projection methods, cocoercive mappings, variational inequality problem

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 22
Issue: 1