IJPAM: Volume 11, No. 2 (2004)

AUXILIARY MINIMIZATION PROBLEM
PRINCIPLE AND ITS APPLICATIONS TO
NONLINEAR VARIATIONAL PROBLEMS

Ram U. Verma
Department of Mathematics
University of Toledo
Toledo, Ohio 43606, USA
e-mail: Verma99@msn.com


Abstract.This paper deals with a crucial role of the partial relaxed monotonicity - a relatively new and less explored concept - in the approximation solvability of a class of nonlinear variational problems on Hilbert spaces, though the convergence analysis much depends on the general auxiliary minimization problem principle. The nonlinear variational inequality problem (NVIP) is stated as follows: find an element $x^{\ast} \in K$ such that $u^{\ast} \in T(x^{\ast })$ and

\begin{displaymath} \langle u^{\ast }, x {-} x^{\ast } \rangle + f(x) {-} f(x^{\ast }) { \ge } 0\,,\quad {\forall } x { \in }K, \end{displaymath}

where $T: K{ \to } P(E^{\ast })$ is a mapping from a nonempty closed convex subset $K$ of a real reflexive Banach space $E$ into the power set $P(E^{\ast })$ of $E^{\ast }$, and $f: K{ \to } R$ is a continuous convex functional on $K$. Here $\langle x^{\ast }, x \rangle $ denotes the duality pairing between elements $x^{\ast }{\in }E^{\ast }$ and elements $x{ \in }E$.

Received: October 11, 2003

AMS Subject Classification: 49J40, 65B05

Key Words and Phrases: auxiliary problem principle, approximation solvability, approximate solutions, partially relaxed monotone mappings, cocoercive mappings

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 11
Issue: 2