* 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 .
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 inequality
In Section 3, we demonstrated the applicability of the abstract results to prove existence of solution(s) for variational inequality
The necessary and sufficient optimality conditions for (6) and (7) can be expressed as the variational inequality
Following 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 , Wangkeeree and Preechasilp  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.