Strong Convergence Theorems for Variational Inequalities and Split Equality Problem. (23rd October 2013) - Wu, Yu Jing; Chen, Ru Dong; Shi, Luo Yi
Abstract : Let H 1, H 2, and H 3 be real Hilbert spaces, let C ⊆ H 1, Q ⊆ H 2 be two nonempty closed convex sets, and let A : H 1 → H 3, B : H 2 → H 3 be two bounded linear operators. The split equality problem (SEP) is to find x ∈ C, y ∈ Q such that A x = B y . Let H = H 1 × H 2 ; consider f : H → H a contraction with coefficient 0 < α < 1, a strongly positive linear bounded operator T : H → H with coefficient γ ̅ > 0, and M : H → H is a β -inverse strongly monotone mapping. Let 0 < γ < γ ̅ / α, S = C × Q and G : H → H 3 be defined by restricting to H 1 is A and restricting to H 2 is - B, that is, G has the matrix form G = [ A, - B ] . It is proved that the sequence w n = ( x n, y n ) ⊆ H generated by the iterative method w n + 1 = P S [ α n γ f ( w n ) + ( I - α n T ) P S ( I - γ n G * G ) P S ( w n - λ n M w n ) ] converges strongly to w ̃ which solves the SEP and the following variational inequality: 〈 ( T - λ f ) w ̃, w - w ̃ 〉 ≥ 0 and 〈 M w ̃, w - w ̃ 〉 ≥ 0 for all w ∈ S . Moreover, if we take M = G * G : H → H, γ n = 0, then M is a β -inverse strongly monotone mapping, and the sequence w n generated by the iterative method w n + 1 = α n γ f ( w n ) + ( I - α n T ) P S ( w n - λ n G * G w n ) converges strongly to w ̃ which solves the SEP and the following variational inequality: 〈 ( T - λ f ) w ̃, w - w ̃ 〉 ≥ 0 for all w ∈ S .