We study the split common solution problem with multiple output sets for monotone operator equations in Hilbert spaces. To solve this problem, we propose two new parallel algorithms. We establish a weak convergence theorem for the first and a strong convergence theorem for the second.
Truong Minh TUYEN1
,
*
,
Nguyen Thi TRANG2
,
Tran Thi HUONG3
. TWO PARALLEL ALGORITHMS FOR A CLASS OF SPLIT COMMON SOLUTION PROBLEMS[J]. Acta mathematica scientia, Series B, 2026
, 46(1)
: 505
-518
.
DOI: 10.1007/s10473-026-0125-2
[1] Bauschke H H, Combettes P L.Convex Analysis and Monotone Operator Theory in Hilbert Spaces. New York: Springer, 2011
[2] Byrne C. Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl, 2002, ${\bf 18}$: 441-453
[3] Byrne C. A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl, 2004, ${\bf 18}$: 103-120
[4] Censor Y, Bortfeld T, Martin B, Trofimov A. A unified approach for inversion problems in intensity modulated radiation therapy. Phys Med Biol, 2006, ${\bf 51}$: 2353-2365
[5] Censor Y, Elfving T. A multiprojection algorithm using Bregman projections in a product space. Numer Algorithms, 1994, ${\bf 8}$: 221-239
[6] Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry,Nonexpansive Mappings. New York: Marcel Dekker, 1984
[7] Halpern B. Fixed points of nonexpanding maps. Bull Amer Math Soc, 1967, ${\bf 73}$: 591-597
[8] Moudafi A. Viscosity approximation methods for fixed-points problems. J Math Anal Appl, 2000, ${\bf 241}$: 46-55
[9] Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Amer Math Soc, 1967, ${\bf 73}$: 591-597
[10] Reich S, Tuyen T M, Mai T N H. The split feasibility problem with multiple output sets in Hilbert spaces. Optim Lett, 2020, ${\bf 14}$: 2335-2353
[11] Reich S, Tuyen T M, Mai T N H. An optimization approach to solving the split feasibility problem in Hilbert spaces. J Glob Optim, 2021, ${\bf 79}$: 837-852
[12] Reich S, Tuyen T M, Trang N T. New iterative algorithms for solving a class of split common solution problems and their applications. Journal of Computational and Applied Mathematics, 2024, ${\bf 440}$: 115637
[13] Rockafellar R. Monotone operators and the proximal point algorithm. SIAM J Control Optim, 1976, ${\bf 14}$: 877-898
[14] Saejung S, Yotkaew P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal: Theory, Methods & Applications, 2012, ${\bf 75}$: 742-750
