General Iterative Algorithms for Hierarchical Fixed Points Approach to Variational Inequalities

This paper deals with new methods for approximating a solution to the fixed point problem; find x ̃ ∈ F ( T ) , where H is a Hilbert space, C is a closed convex subset of H , f is a ρ -contraction from C into H , 0 < ρ < 1 , A is a strongly positive linear-bounded operator with coefficient γ ̅ > 0 , 0 < γ < γ ̅ / ρ , T is a nonexpansive mapping on C, and P F ( T ) denotes the metric projection on the set of fixed point of T . Under a suitable different parameter, we obtain strong convergence theorems by using the projection method which solves the variational inequality 〈 ( A - γ f ) x ̃ + τ ( I - S ) x ̃ , x - x ̃ 〉 ≥ 0 for x ∈ F ( T ) , where τ ∈ [ 0 , ∞ ) . Our results generalize and improve the corresponding results of Yao et al. (2010) and some authors. Furthermore, we give an example which supports our main theorem in the last part.