Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

We consider a general variational inequality and fixed point problem, which is to find a point x(*) with the property that (GVF): x(*) is an element of GVI(C,A) and g(x(*)) is an element of Fix(S) where GVI(C,A) is the solution set of some variational inequality Fix(S) is the fixed points set of nonexpansive mapping S, and g is a nonlinear operator. Assume the solution set Omega of (GVF) is nonempty. For solving (GVF), we suggest the following method g(x(n+1)) = beta g(x(n)) + (1 - beta)SPC [alpha F-n(x(n)) + (1 - alpha(n)) (g(x(n)) - lambda Ax(n))], n >= 0. It is shown that the sequence {x(n)} converges strongly to x(*) is an element of Omega which is the unique solution of the variational inequality < F(x(*)) - g(x(*)), g(x) - g (x(*))> <= 0, for all x is an element of Omega.