Convergence theorems of solutions of a generalized variational inequality

Abstract

The convex feasibility problem (CFP) of finding a point in the nonempty intersection ∩m=1r Cm is considered, where r ≥ 1 is an integer and each Cm is assumed to be the solution set of a generalized variational inequality. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let Am, Bm : C → H be relaxed cocoercive mappings for each 1 ≤ m ≤ r. It is proved that the sequence {xn} generated in the following algorithm: x1 ∈ C, xn+1 = αnu + βnx n + γn ∑m=1r δ(m,n)PC(τmBmx n - λmAmxn), n ≥ 1, where u ∈ C is a fixed point, {αn}, {βn}, {γn}, {δ(1,n)}, ..., and {δ (r,n)} are sequences in (0, 1) and {τm} m=1r, {λm}m=1r are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions. © 2011 Yu and Liang; licensee Springer.