Abstract
The approximation-solvability of the following nonlinear variational inequality (NVI) problem is presented:
Determine an element x∗∈K such that
〈T(x∗),x−x∗〉≥0
for all
x∈K,
where T:K→H is a mapping from a nonempty closed convex subset K of a real Hilbert space H into H. The iterative procedure is characterized as a nonlinear variational inequality (for any arbitrarily chosen initial point x0∈K)
〈ρT(PK[xk−ρT(xk)])+xk+1−xk,x−xk+1〉≥0for all
x∈K
and for
k≥0,
which is equivalent to a double projection formula
xk+1=PK[xk−ρT(PK[xk−ρT(xk)])],
where PK denotes the projection of H onto K.