Volume 69 | Issue 2 | Year 2023 | Article Id. IJMTT-V69I2P517 | DOI : https://doi.org/10.14445/22315373/IJMTT-V69I2P517
Received | Revised | Accepted | Published |
---|---|---|---|
10 Jan 2023 | 12 Feb 2023 | 22 Feb 2023 | 28 Feb 2023 |
In the paper, a new shrinking projection method is proposed for solving split equality problem(SEP) in Banach spaces. For practical purposes, we substitute duality mapping for inner in Banach spaces. Under proper conditions, we give proofs of strong convergence for the SEP with two different choices of the step-size. Finally, we make some extensions and generalization.
[1] Hedy Attouch et al., “Alternating Proximal Algorithms for Weakly Coupled Minimization Problems, Applications to Dynamical Games and PDE’s,” Journal of Convex Analysis, vol. 15, no. 3, pp. 485-506, 2008.
[2] Yasir Arfat, “Multi-Inertial Parallel Hybrid Projection Algorithm for Generalized Split Null Point Problems,” Journal of Applied Mathematics and Computing, vol. 68, pp. 3179-3198, 2022. Crossref, https://doi.org/10.1007/s12190-021-01660-4
[3] Yair Censor, Tommy Elfving, “A Multiprojection Algorithm Using Bregman Projections in a Product Space,” Numerical Algorithms, vol. 8, pp. 221-239, 1994. Crossref, https://doi.org/10.1007/BF02142692
[4] Yair Censor et al., “The Multiple-Sets Split Feasibility Problem and its Applications for Inverse Problems,” Inverse Problems, vol. 21, no. 6, pp. 2071-2084, 2005. Crossref, https://doi.org/10.1088/0266-5611/21/6/017
[5] Yair Censor et al., “A Unified Approach for Inversion Problems in Intensity-Modulated Radiation Therapy,” Physics in Medicine & Biology, vol. 51, no. 10, pp. 2353-2365, 2006. Crossref, https://doi.org/10.1088/0031-9155/51/10/001
[6] J. Lindenstrauss, and L. Tzafriri , Classical Banach spaces II: Function spaces, Springer, 1979.
[7] Ioana Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Springer Netherlands, 1990.
[8] Joseph Diestel, Geometry of Banach Space-Selected Topics, Springer, 1975.
[9] Mohammad Eslamian, Yekini Shehu and Olaniyi S. Iyiola, “A Strong Convergence Theorem for a General Split Equality Problem with Applications to Optimization and Equilibrium Problem,” Calcolo, vol. 55, no. 4, pp. 1-31, 2018. Crossref, https://doi.org/10.1007/s10092- 018-0290-3
[10] T. Figiel, “On the Moduli of Convexity and Smoothness,” Studia Mathematica, vol. 56, no. 2, pp. 12-55, 1976.
[11] O.S. Iyiola, and Y. Shehu, “A Cyclic Iterative Method for Solving Multiple Sets Split Feasibility Problems in Banach Spaces,” Quaestiones Mathematicae, vol. 39, no. 7, pp. 959-975, 2016. Crossref, https://doi.org/10.2989/16073606.2016.1241957
[12] Abdellatif Moudafi, “A Relaxed Alternating CQ-Algorithm for Convex Feasibility Problems.,” Nonlinear Analysis: Theory, Methods & Applications, vol. 79, pp. 117-121, 2013. Crossref, https://doi.org/10.1016/j.na.2012.11.013
[13] F Schöpfer, Thomas Schuster, and Alfred K. Louis, “ An Iterative Regularization Method for the Solution of the Split Feasibility Problem in Banach Spaces,” Inverse Problems, vol. 24, no. 5, 2008. Crossref, https://doi.org/10.1088/0266-5611/24/5/055008
[14] Dianlu Tian, and Lining Jiang, “Two-Step Methods and Relaxed Two-Step Methods for Solving the Split Equality Problem,” Computational and Applied Mathematics, vol. 40, pp. 83, 2021. Crossref, https://doi.org/10.1007/s40314-021-01465-y
[15] F. Schöpfer, A.K. Louis, and T. Schuster, “Nonlinear Iterative Methods for Linear ill-Posed Problems in Banach Spaces,” Inverse Problems, vol. 22, no. 1, pp.311, 2006. Crossref, https://doi.org/10.1088/0266-5611/22/1/017
[16] Luo Yi Shi, Rudong Chen, and Yujing Wu, “Strong Convergence of Iterative Algorithms for Split Equality Problem,” Journal of Inequalities and Applications, vol. 2014, no. 478, 2014. Crossref, https://doi.org/10.1186/1029-242X-2014-478
[17] Suthep Suantai, “Strong Convergence of a Self-Adaptive Method for the Split Feasibility Problem in Banach Spaces,” Journal of Fixed Point Theory and Applications, vol. 20, no. 68, pp. 1-21, 2018. Crossref, https://doi.org/10.1007/s11784-018-0549-y
[18] Yekine Shehu, and Olaniyi S. Iyiola, “Strong Convergence Result for Proximal Split Feasibility Problem in Hilbert Spaces,” Optimization, vol. 66, no. 12, pp. 2275-2290, 2017. Crossref, https://doi.org/10.1080/02331934.2017.1370648
[19] Yekine Shehu, and Olaniyi S. Iyiola, “Convergence Analysis for the Proximal Split Feasibility Problem Using an Inertial Extrapolation Term Method,” Journal of Fixed Point Theory and Applications, vol. 19, pp. 2483-2510, 2017. Crossref, https://doi.org/10.1007/s11784- 017-0435-z Ziyuan Zhang & Tongxin Xu / IJMTT, 69(2), 147-154, 2023 154
[20] Yekine Shehu, Ferdinard Ogbuisi, and Olaniyi S. Iyiola, “Strong Convergence Theorem for Solving Split Equality Fixed Point Problem which Does not Involve the Prior Knowledge of Operator Norms,” Bulletin of the Iranian Mathematical Society, vol. 43, no. 2, pp. 349- 371, 2017.
[21] Y.Shehu, O.S. Iyiola, and C.D. Enyi, “An Iterative Algorithm for Solving Split Feasibility Problems and Fixed Point Problems in Banach Spaces,” Numerical Algorithms, vol. 72, no. 4, pp. 835-864, 2016. Crossref, https://doi.org/10.1007/s11075-015-0069-4
[22] Wataru Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, 2000.
[23] Wataru Takahashi, “The Split Feasibility Problem in Banach Spaces,” Journal of nonlinear and convex analysis, vol. 15, no. 6, pp. 1349- 1355, 2014.
[24] Wataru Takahashi, “The Split Feasibility Problem and the Shrinking Projection Method in Banach Spaces,” Journal of Nonlinear and Convex Analysis, vol. 16, no. 7, pp. 1449-1459, 2015.
Ziyuan Zhang, Tongxin Xu, "A Shrinking Projection Algorithm for Solving Split Equality Problem in Banach Spaces," International Journal of Mathematics Trends and Technology (IJMTT), vol. 69, no. 2, pp. 147-154, 2023. Crossref, https://doi.org/10.14445/22315373/IJMTT-V69I2P517