Shrinking projection method for proximal split feasibility and fixed point problems

Yıl 2019, Cilt: 2 Sayı: 2, 92 - 101, 30.08.2019

Jinzuo Chen

Öz

In this paper, we consider and study proximal split feasibility and fixed point problem. For solving the problems, we introduce an iterative algorithm with shrinking projection technique. It is proven that the sequence generated by the proposed iterative algorithm converges strongly to the common solution of the proximal split feasibility and fixed point problems.

Anahtar Kelimeler

shrinking projection method, proximal split feasibility problem, fixed point problem, k-strictly pseudo-contractive mapping

Kaynakça

• [1] Censor, Y, Elfving, T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithm 8,221-239(1994)
• [2] Byrne, C: A unified treatment of some iterative algorithm in signal processing and image reconstruction. Inverse Probl. 20,103-120(2004)
• [3] Censor, Y, Motova, A, Segal, A: Perturbed projections and subgradient projections for the multiple-sets split feasibilityproblem. J. Math. Anal. Appl. 327, 1244-1256(2007)
• [4] Byrne, C: Iterative oblique projection onto convex subsets and the split feasibility problem, Inverse Probl. 18, 441-453(2002)
• [5] Yao, Y., Liou, Y., Yao, J: Iterative algorithms for the split variational inequality and fixed point problems under nonlineartransformations. J. Nonlinear Sci. Appl., 10, 843–854(2017).
• [6] Yao, Y., Leng, L., Postolache, M., Zheng, X: Mann-type iteration method for solving the split common fixed point problem.J. Nonlinear Convex Anal., 18, 875–882(2017).
• [7] Censor, Y. Zaknoon, M: Algorithms and convergence results of projection methods for inconsistent feasibility problems: Areview. Pure and Applied Functional Analysis, 3, 565-586(2018).
• [8] Ceng, LC, Ansari, QH, Yao, JC: Relaxed extragradient methods for finding minimum-norm solutions of the split feasibilityproblem. Nonlinear. Anal. 75, 2116-2125(2012)
• [9] Yao, Y., Yao, J., Liou, Y., Postolache, M: Iterative algorithms for split common fixed points of demicontractive operatorswithout priori knowledge of operator norms. Carpathian J. Math., 34(3), 459–466(2018).
• [10] Chen, P. He, H. Liou, Y. Wen, C: Convergence rate of the CQ algorithm for split feasibility problems. J. Nonlinear ConvexAnal., 19(3), 381-395(2018).
• [11] Padcharoen, A. Kumam, P, Cho, Y. Thounthong, P: A Modified Iterative Algorithm for Split Feasibility Problems of RightBregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications. Algorithm. 9(4), 75 (2016).
• [12] Chen, JZ, Ceng, LC, Qiu, YQ, Kong, ZR: Extra-gradient methods for solving split feasibility and fixed point problems.Fixed Point Theory Appl. 2015, Article ID 192(2015)
• [13] Lopez, G, Martin-Marquez, V, Wang, F, Xu, HK: Solving the split feasibility problem without prior knowledge of matrixnorms. Inverse Probl. 28, 085004(2012)
• [14] Moudafi, A, Thakur, BS: Solving proximal split feasibility problems without prior knowledge of operator norms. Optim.Lett. 8, 2099-2110(2014)
• [15] Takahashi, W, Takeuchi, Y, Kubota, R: Strong convergence theorems by hybrid methods for families of nonexpansivemappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276-286(2008)
• [16] Marino, G, Xu, HK: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math.Anal. Appl. 329, 336-346(2007)
• [17] Yao, Liou, Y, Kang, S: Approach to common elements of variational inequality problems and fixed point problems via arelaxed extragradient method. Computers Math. Appl. 59, 3472-3480(2010)
• [18] Martinez-Yanes, Xu, HK: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear. Anal. 64,2400-2411(2006)
• [19] Yao, Y, Yao, Z, Abdou, AA, Cho, YJ: Self-adaptive algorithms for proximal split feasibility problems and strong convergenceanalysis. Fixed Point Theory Appl. 2015, Article ID 205(2015)
• [20] Yao, Z, Cho, SY, Kang, SM, Zhu, LJ: A Regularized Algorithm for the Proximal Split Feasibility Problem. Abstr. Appl.Anal. 2014, Article ID 894272(2014)
• [21] Shehu Y, Ogbuisi FU: Convergence analysis for proximal split feasibility problems and fixed point problems. J. Appl. Math.Comput. 48, 221-239(2015)