DIFFERENtIAl & DIFFERENCE EqUAtIONS ANd APPlICAtIONS

NONLINEAR VARIATIONAL INCLUSION PROBLEMS
INVOLVING A-MONOTONE MAPPINGS
RAM U. VERMA
Based on the notion of A-monotonicity, a new class of nonlinear variational inclusion
problems is introduced. Since A-monotonicity generalizes H-monotonicity (and in turn
generalizes maximal monotonicity), results thus obtained are general in nature.
Copyright © 2006 Ram U. Verma. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Resolvent operator techniques have been in literature for a while for solving problems
from several fields, including complementarity, optimization, mathematical programming,
equilibria in economics, and variational inclusions, but the generalized resolvent
operator technique (referred to as A-resolvent operator technique) based on A-monotonicity
[13, 14] is a new development. This gave rise to several generalized resolvent
operator-like techniques that can be applied to several variational inclusion problems
from sensitivity analysis, model equilibria problems in economics, and optimization and
control theory. Just recently, the author [13, 14] generalized the notion of the maximal
monotonicity to A-monotonicity, andappliedA-resolvent operator technique, thus developed,
to establishing existence and uniqueness of the solution as well as algorithmic
convergence analysis for the solution of nonlinear variational inclusions.
We explore in this paper the role of A-monotonicity in constructing a general framework
for A-resolvent operator technique, and then we consider the existence and uniqueness
of the solution and convergence analysis for approximate solution of a new class
of nonlinear variational inclusion problems involving relaxed cocoercive mappings using
A-resolvent operator technique. As there is a vast literature on variational inequalities
and their applications to several fields of research, the obtained nonlinear variational
inclusion results generalize the recent research works of Fang and Huang [3, 4], Liu et
al. [9], and Jin [7] to the case of A-monotone mappings. For more details, we refer to
[1–14].
Hindawi Publishing Corporation
Proceedings of the Conference on Differential & Difference Equations and Applications, pp. 1067–1076