Convergence theorems of solutions of a generalized variational inequality

Abstract

Abstract The convex feasibility problem (CFP) of finding a point in the nonempty intersection 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: where u ∈ C is a fixed point, {αn}, {βn}, {γn}, {δ(1,n)}, ..., and {δ(r,n)} are sequences in (0, 1) and , are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions. 2000 AMS Subject Classification: 47H05; 47H09; 47H10.

Keywords

• nonexpansive mapping
• fixed point
• relaxed cocoercive mapping
• variational inequality