Abstract and Applied Analysis

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 $f(x,y)=(1/2){∥Ax-By∥}_{2}^{2}$ 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).

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 813635, 5 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393444394

Digital Object Identifier
doi:10.1155/2013/813635

Mathematical Reviews number (MathSciNet)
MR3049423

Zentralblatt MATH identifier
1278.90442

Citation

Chen, Rudong; Li, Junlei; Ren, Yijie. Regularization Method for the Approximate Split Equality Problem in Infinite-Dimensional Hilbert Spaces. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 813635, 5 pages. doi:10.1155/2013/813635. https://projecteuclid.org/euclid.aaa/1393444394

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