A continuous semigroup , , in a Hilbert space is called nonexpansive if for arbitrary in . For a bifunction , denotes the set of equilibrium points, i.e., all such that for all . The present paper studies iterative methods of finding solutions to a variational problem among equilibrium points, which are fixed points of .
The authors present two iterative constructions, which they call implicit and explicit. In the implicit iterative construction, a pair of continuous sequences is defined as solution to
Here, and is a strongly positive bounded linear operator.
The explicit iterations are defined by the system
It is demonstrated that, with an appropriate definition of the coefficients , the implicit sequences , and the the explicit sequences , , both converge strongly to the unique solution to variational inequality
on the set of fixed points of the semigroup belonging to .
No examples illustrate the theoretical results.