2]) [member of] C(0, T; V x [gamma]) and g [member of] C(0, T; V) be given and consider the following

variational inequality.

Variational inequality theory plays an important role in many fields, such as optimal control, mechanics, economics, transportation equilibrium, and engineering science.

The development of

variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research.

Yamada: The hybrid steepest-descent method for

variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings, Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, Edited by D.

is said to be the

variational inequality associated with X and [PHI], denoted by VI(X, [PHI]).

being incorporated in a

variational inequality, we can still derive (3.

Liu, An inverse coefficient problemfor for a nonlinear parabolic

variational inequality, Appl.

Recall that the classical

variational inequality problem, denoted by VI(C,A), is to find u [member of] C such that

The combined models can be formulated by using the equivalent optimization approach (Florian and Nguyen 1978; Safwat and Magnanti 1988; Lam and Huang 1992; Abrahamsson and Lundqvist 1999),

variational inequality (VI) approach (Dafermos 1982; Florian et al.

The innovative and novel feature of a projected dynamical system is that its set of stationary points corresponds to the set of solutions of the corresponding

variational inequality problems.

This volume addresses research on a number of fields, including fixed-point theory, convex and set-valued analysis,

variational inequality, and complementary problem theory, nonlinear ergodic theory, difference, differential and integral equations, control and optimization theory, dynamic system theory, inequality theory, stochastic analysis and probability theory, and their applications.

It is well known that the theory of

variational inequality has been developed as a class of important tools for the study of minimization problems; see, for example, [13].