Strong convergence of iterative algorithms for the split equality problem

Abstract

Abstract Let H 1 , H 2 , H 3 be real Hilbert spaces, 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 finding x ∈ C , y ∈ Q such that A x = B y . Recently, Moudafi has presented the ACQA algorithm and the RACQA algorithm to solve SEP. However, the two algorithms are weakly convergent. It is therefore the aim of this paper to construct new algorithms for SEP so that strong convergence is guaranteed. Firstly, we define the concept of the minimal norm solution of SEP. Using Tychonov regularization, we introduce two methods to get such a minimal norm solution. And then, we introduce two algorithms which are viewed as modifications of Moudafi’s ACQA, RACQA algorithms and KM-CQ algorithm, respectively, and converge strongly to a solution of SEP. More importantly, the modifications of Moudafi’s ACQA, RACQA algorithms converge strongly to the minimal norm solution of SEP. At last, we introduce some other algorithms which converge strongly to a solution of SEP.