The purpose of this paper is to introduce iterative method for solving generalized split inverse problem given as a task of finding a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. The proposed algorithm is formulated in low-cost sequential computing method with step size selection technique. The strong convergence theorem of the proposed algorithm is derived under the appropriate assumptions. The result presented in the paper generalized several results in the literature. Numerical example is given to illustrate the efficiency and performance of the proposed iterative method.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Browder FE, Petryshyn WV (1967) Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 20:197–228
Byrne C (2003) A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl 20:103
Byrne C, Censor Y, Gibali A, Reich S (2012) The split common null point problem. J Nonlinear Convex Anal 13:759775
Cegielski A (2015) General method for solving the split common fixed point problem. J Optim Theory Appl 165:385404
Censor Y, Elfving T (1994) A multiprojection algorithm using Bregman projections in a product space. Numer Algorithm 8:221–239
Censor Y, Segal A (2009) The split common fixed point problem for directed operators. J Convex Anal 16:587–600
Censor Y, Elfving T, Kopf N, Bortfeld T (2005) The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl 21:2071
Censor Y, Chen W, Combettes PL, Davidi R, Herman GT (2012a) On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput Optim Appl 51:1065–1088
Censor Y, Gibali A, Reich S (2012b) Algorithms for the split variational inequality problem. Numer Algorithms 59:301–323
Combettes PL (1996) The convex feasibility problem in image recovery. In: Advances in imaging and electron physics, vol 95. Elsevier, pp 155–270
Dautray R, Lions JL (2012) Mathematical analysis and numerical methods for science and technology: volume 6 evolution problems II. Springer Science & Business Media, New York
Eslamian M (2016) General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65:443–465
Fattorini HO, Fattorini HO (1999) Infinite dimensional optimization and control theory, vol 54. Cambridge University Press, Cambridge
Gebrie AG, Wangkeeree R (2019) Proximal method of solving split system of minimization problem. J Appl Math Comput 63:1–26
Gebrie AG, Wangkeeree R (2020) Parallel proximal method of solving split system of fixed point set constraint minimization problems. RACSAM 114:13
He Z (2012) The split equilibrium problem and its convergence algorithms. J Inequal Appl 2012:162
Hendrickx JM, Olshevsky A (2010) Matrix p-norms are NP-hard to approximate if \(p 6= 1,2,\infty \). SIAM J Matrix Anal Appl 31:2802–2812
Maingé PE (2008) Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal 16:899–912
Marino G, Xu HK (2007) Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 329:336–46
Moudafi A (2010) The split common fixed-point problem for demicontractive mappings. Inverse Probl 26:055007
Moudafi A (2011a) A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal 74:4083–4087
Moudafi A (2011b) Split monotone variational inclusions. J Optim Theory Appl 150:275–283
Suparatulatorn R, Suantai S, Phudolsitthiphat N (2019) Reckoning solution of split common fixed point problems by using inertial self-adaptive algorithms. RACSAM 113:3101–3114
Takahashi W (2017) The split common fixed point problem and the shrinking projection method in Banach spaces. J Convex Anal 24:1015–1028
Tang YC, Liu LW (2016) Several iterative algorithms for solving the split common fixed point problem of directed operators with applications. Optimization 65:53–65
Wang F (2017a) A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66:407–415
Wang F (2017b) A new method for split common fixed-point problem without priori knowledge of operator norms. J Fixed Point Theory Appl 19:2427–2436
Wang F, Xu HK (2011) Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal 74:4105–4111
Xu HK (2002) Iterative algorithms for nonlinear operators. J Lond Math Soc 66:240–56
Yao Y, Liou YC, Postolache M (2018) Self-adaptive algorithms for the split problem of the demicontractive operators. Optimization 67:1309–1319
Zhao J, Yang Q (2013) A simple projection method for solving the multiple-sets split feasibility problem. Inverse Probl Sci Eng 21:537–546
Zhao J, Hou D (2019) A self-adaptive iterative algorithm for the split common fixed point problems. Numer Algorithms 82:1047–1063
The author would like to thank the Associate Editor and anonymous reviewers for their comments and suggestions on improving an earlier version of this paper.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Communicated by Natasa Krejic.
About this article
Cite this article
Gebrie, A.G. A novel low-cost method for generalized split inverse problem of finite family of demimetric mappings. Comp. Appl. Math. 40, 40 (2021). https://doi.org/10.1007/s40314-021-01437-2
- Split inverse problem
- Demimetric mapping
- Fixed point
- Self-adaptive technique
Mathematics Subject Classification