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

Abstract

We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let $H_1$, $H_2$, and $H_3$ be infinite-dimensional real Hilbert spaces, let $C\subset H_1$ and $Q\subset H_2$ be two nonempty closed convex sets, and let $A:H_1\to H_3$ and $B:H_2\to H_3$ be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function $f\left(x,y\right)=\mathrm{(1}/\mathrm{2)}{∥Ax-By∥}_2^2$ over $x\in C$ and $y\in Q$. 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).