Generalized auxiliary problem principle and solvability of a class of nonlinear variational inequalities involving cocoercive and co-Lipschitzian mappings

Volume 2, Issue 3, 2001

Article 27

GENERALIZED AUXILIARY PROBLEM PRINCIPLE AND SOLVABILITY OF A CLASS OF NONLINEAR VARIATIONAL INEQUALITIES INVOLVING COCOERCIVE AND CO-LIPSCHITZIAN MAPPINGS

RAM U. VERMA

UNIVERSITY OF TOLEDO
DEPARTMENT OF MATHEMATICS
TOLEDO, OHIO 43606, USA
E-Mail: verma99@msn.com

Received _____; accepted 15 March, 2001.
Communicated by: D. Bainov

ABSTRACT. The approximation-solvability of the following class of nonlinear variational inequality (NVI) problems, based on a new generalized auxiliary problem principle, is discussed.

Find an element $x^{\ast }\in$ such that

$\displaystyle \left\langle (S-T)\left( x^{\ast }\right) ,x-x^{\ast }\right\rangle +f(x)-f\left( x^{\ast }\right) \geq 0$ for all $\displaystyle x\in K,$

where $S,T:K\rightarrow H$ are mappings from a nonempty closed convex subset

of a real Hilbert space

into

, and $f:K\rightarrow \mathbb{R}$ is a continuous convex functional on

The generalized auxiliary problem principle is described as follows: for given iterate $x^{k}\in K$ and, for constants $\rho >0$ and $\sigma >0$ ), find $x^{k+1}$ such that

	$\displaystyle \left\langle \rho (S-T)\left( y^{k}\right) +h^{\prime }\left( x^{k+1}\right) -h^{\prime }\left( y^{k}\right) ,x-x^{k+1}\right\rangle$
	$\displaystyle \qquad\qquad +\rho (f(x)-f(x^{k+1}))\geq 0$ for all $\displaystyle x\in K,$

where

	$\displaystyle \left\langle \sigma (S-T)\left( x^{k}\right) +h^{\prime }\left( y^{k}\right) -h^{\prime }\left( x^{k}\right) ,x-y^{k}\right\rangle$
	$\displaystyle \qquad\qquad +\sigma (f(x)-f(y^{k}))\geq 0\,\,$ for all $\displaystyle x\in K,$

where

is a functional on

and $h^{\prime }$ the derivative of

Key words:
Generalized auxiliary variational inequality problem, Cocoercive mappings, Approximation-solvability, Approximate solutions, Partially relaxed monotone mappings.

2000 Mathematics Subject Classification:
49J40.

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