Regularization Method for the Approximate Split Equality Problem in Infinite‐Dimensional Hilbert Spaces

We studied the approximate split equality problem (ASEP) in the framework of infinite‐dimensional Hilbert spaces. Let H1, H2, and H3 be infinite‐dimensional real Hilbert spaces, let C ⊂ H1 and Q ⊂ H2 be two nonempty closed convex sets, and let A : H1 → H3 and B : H2 → H3 be two bounded linear operators. The ASEP in infinite‐dimensional Hilbert spaces is to minimize the function fx,y=(12)/Ax-By22 over x ∈ C and y ∈ Q. Recently, Moudafi and Byrne had proposed several algorithms for solving the split equality problem and proved their convergence. Note that their algorithms have only weak convergence in infinite‐dimensional Hilbert spaces. In this paper, we used the regularization method to establish a single‐step iterative for solving the ASEP in infinite‐dimensional Hilbert spaces and showed that the sequence generated by such algorithm strongly converges to the minimum‐norm solution of the ASEP. Note that, by taking B = I in the ASEP, we recover the approximate split feasibility problem (ASFP).