Iterative Algorithms Approach to Variational Inequalities and Fixed Point Problems

Abstract

We consider a general variational inequality and fixed point problem, which is to find a point x* with the property that (GVF): x*∈GVI(C,A) and g(x*)∈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 Ω of (GVF) is nonempty. For solving (GVF), we suggest the following method g(xn+1)=βg(xn)+(1-β)SPC[αnF(xn)+(1-αn)(g(xn)-λAxn)], n≥0. It is shown that the sequence {xn} converges strongly to x*∈Ω which is the unique solution of the variational inequality 〈F(x*)-g(x*),g(x)-g(x*)〉≤0, for all x∈Ω.