* We first formulate the online resource provisioning problem for virtual clusters in the data center network with the objective of maximizing the total revenue by using

variational inequality. We prove the existence and uniqueness of the optimal solution for the online provision problem.

which is called the

variational inequality, introduced and studied by Stampacchia [23].

We consider the following

variational inequality problem (VI(A, C)): find a u [member of] C such that

Since K is a nonempty closed convex subset of the Hilbert space [H.sup.2.sub.0]([OMEGA]), it follows from the standard theory of calculus of variations [20, 25] that the obstacle problem (1.1) has a unique solution u [member of] K characterized by the fourth-order

variational inequalityIn Section 3, we demonstrated the applicability of the abstract results to prove existence of solution(s) for

variational inequality problems.

The necessary and sufficient optimality conditions for (6) and (7) can be expressed as the

variational inequalityFollowing an approach adopted in [4,7], we formulate the boundary-value problem (1)-(9) as a

variational inequality for the unknown velocity field.

Afterwards, based on the concept of quasilinearization introduced by Berg and Nikolaev [15], Wangkeeree and Preechasilp [6] explored strong convergence results of (4) and (5) in CAT(0) spaces without the property P and presented that the iterative processes (4) and (5) converges strongly to [??] [member of] Fix(g) such that [mathematical expression not reproducible] is the unique solution of the following

variational inequality:

As the chains enter the dynamic competition, we use

variational inequality formulation to find equilibrium results.

From the main result, we deduce the Himmelberg type fixed point theorem for acyclic compact multifunctions, acyclic versions of general geometric properties of convex sets, abstract

variational inequality theorems, new minimax theorems, and non-continuous versions of the Brouwer and Kakutani type fixed point theorems with very generous boundary conditions.

Assuming the users' route choice behavior follows the General SUE principle; this article proposed a trial-and-error method based on the

variational inequality (VI) model built by Liu et al.

Let [epsilon] [right arrow] 0; then

variational inequality (36) can be obtained.