Strong Convergence Theorems for Variational Inequalities and Split Equality Problem

Abstract EN

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 Ax=By.

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 γ̅>0, and M:H→H is a β-inverse strongly monotone mapping.

Let 0<γ<γ̅/α, S=C×Q and G:H→H3 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 {wn}={(xn,yn)}⊆H generated by the iterative method wn+1=PS[αnγf(wn)+(I-αnT)PS(I-γnG*G)PS(wn-λnMwn)] converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 and 〈Mw̃,w-w̃〉≥0 for all w∈S.

Moreover, if we take M=G*G:H→H, γn=0, then M is a β-inverse strongly monotone mapping, and the sequence {wn} generated by the iterative method wn+1=αnγf(wn)+(I-αnT)PS(wn-λnG*Gwn) converges strongly to w̃ which solves the SEP and the following variational inequality: 〈(T-λf)w̃,w-w̃〉≥0 for all w∈S.