Strong convergence theorems for variational inequalities and split equality problem

Let h1,h2,and h3 be real hilbert spaces,let c h1,q h2 be two nonempty closed convex sets,and let a: h1 → h3,b: h2 → h3 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 = h1 × h2; 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 h1 is a and restricting to h2 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 - n m w n) ] converges strongly to w which solves the sep and the following variational inequality: 〈 (t - 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 - n g g w n) converges strongly to w which solves the sep and the following variational inequality: 〈 (t - f) w,w - w 〉 ≥ 0 for all w s. © 2013 yu jing wu et al.