Abstract
We studied the approximate split equality problem (ASEP) in the framework of infinite-dimensional Hilbert spaces. Let , , and   be infinite-dimensional real Hilbert spaces, let and   be two nonempty closed convex sets, and let and   be two bounded linear operators. The ASEP in infinite-dimensional Hilbert spaces is to minimize the function over and . 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 in the ASEP, we recover the approximate split feasibility problem (ASFP).
Highlights
Let C ⊆ RN and Q ⊆ RM be closed, nonempty convex sets, and let A and B be J by N and J by M real matrices, respectively
Assume that an is a sequence of nonnegative real numbers such that an+1 ≤ (1 − γn) an + γnδn, n ≥ 0, (12)
Assume that the minimization (14) is consistent, and let ωmin be its minimum-norm solution; namely, ωmin ∈ Γ (Γ is the solution set of the minimization (14)) has the property
Summary
Let C ⊆ RN and Q ⊆ RM be closed, nonempty convex sets, and let A and B be J by N and J by M real matrices, respectively. In [12], Byrne considered and studied the algorithms to solve the approximate split equality problem (ASEP), which can be regarded as containing the consistent case and the inconsistent case of the SEP There, he proposed a simultaneous iterative algorithm (SSEA) as follows: xk+1 = PC (xk − γkAT (Axk − Byk)) , (4). Yk+1 = PQ (yk + γkBT (Axk − Byk)) , where ε ≤ γk ≤ (2/ρ(GTG)) − ε He proposed the relaxed SSEA (RSSEA) and the perturbed version of the SSEA (PSSEA) for solving the ASEP, and he proved their convergence. He used these algorithms to solve the approximate split feasibility problem (ASFP), which is a special case of the ASEP. We use the regularization method to establish a single-step iterative to solve the ASEP in infinitedimensional Hilbert spaces, and we will prove its strong convergence
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