Abstract
The split common fixed point problem is an optimization challenge that involves finding an element within one fixed point set such that when transformed by a bounded linear operator, it belongs to another fixed point set. This problem falls under the category of inverse problems in mathematics. We present a novel self-adaptive algorithm based on double inertial steps for solving the split common fixed point problem for demicontractive mappings. We also establish a weak convergence theorem for our method. Furthermore, we also present some numerical experiments illustrating the convergence behavior and the efficiency of our proposed algorithm.
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The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Alvarez F, Attouch H (2001) An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal 9:3–11
Bauschke HH, Borwein JM (1996) On projection algorithms for solving convex feasibility problems. SIAM Rev 38:367–426
Bejenaru A, Ciobanescu C (2022) New partially projective algorithm for split feasibility problems with application to BVP. J Nonlinear Convex Anal 23(3):485–500
Byrne C (2002) Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Problems 18:441–453
Byrne C (2004) A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Problems 18:103–120
Censor Y, Segal A (2009) The split common fixed point problem for directed operators. J. Convex Anal. 16:587–600
Cui H, Wang F (2014) Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theory Appl 2014:1–8
Kraikaew R, Saejung S (2014) On split common fixed point problems. J Math Anal Appl 415:513–524
Kumar A, Thakur BS, Postolache M (2024) Dynamic stepsize iteration process for solving split common fixed point problems with applications. Math Comput Simul 218:498–511
Moudafi A (2010) The split common fixed point problem for demicontractive mappings. Inverse Problems 26:055007
Nesterov Y (1983) A method for solving the convex programming problem with convergence rate \(O(\frac{1}{k^2})\). Dokl Akad Nauk SSSR 269:543–547
Opial Z (1967) Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc 73:591–597
Polyak BT (1964) Some methods of speeding up the convergence of iterative methods. Zh Vychisl Mat Mat Fiz 4:1–17
Suparatulatorn R, Charoensawan P, Poochinapan K (2019a) Inertial self-adaptive algorithm for solving split feasible problems with applications to image restoration. Math Methods Appl Sci 42:7268–7284
Suparatulatorn R, Suantai S, Phudolsitthiphat N (2019b) Reckoning solution of split common fixed point problems by using inertial self-adaptive algorithms. RACSAM 113:3101–3114
Suparatulatorn R, Cholamjiak P, Suantai S (2020) Self-adaptive algorithms with inertial effects for solving the split problem of the demicontractive operators. RACSAM 114:40. https://doi.org/10.1007/s13398-019-00737-x
Taiwo A, Mewomo O, Gibali A (2021) Simple strong convergent method for solving split common fixed point problems. J Nonlinear Var Anal 5:777–793
Wang F (2017) A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66:407–415
Wang F, Xu HK (2011) Cyclic algorithms for split feasibility problems in Hilbert spaces. Nonlinear Anal 74:4105–4111
Wang F, Xu HK (2018) Weak and strong convergence of two algorithms for the split fixed point problem. Numer Math Theory Method Appl 11:770–781
Xu HK (2006) A variable Krasnosel’skii–Mann algorithm and the multiple-set split feasibility problem. Inverse Problems 22:2021–2034
Xu HK (2010) Iterative methods for the split feasibility problem in infinite dimensional Hilbert spaces. Inverse Problems 26:105018
Yang Q (2004) The relaxed CQ algorithm for solving the problem split feasibility problem. Inverse Problems 20:1261–1266
Yao Y, Liou YC, Postolache M (2018) Self-adaptive algorithms for the split problem of the demicontractive operators. Optimization 67:1309–1319
Acknowledgements
The authors would like to thank the Editor and the two referees for their valuable comments and suggestions which helped us very much in improving and presenting the original version of this paper. This research is funded by the National Economics University, Hanoi, Vietnam.
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This research was funded by the National Economics University, Hanoi, Vietnam.
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Thong, D.V., Reich, S., Li, XH. et al. An efficient algorithm with double inertial steps for solving split common fixed point problems and an application to signal processing. Comp. Appl. Math. 44, 102 (2025). https://doi.org/10.1007/s40314-024-03058-x
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DOI: https://doi.org/10.1007/s40314-024-03058-x
Keywords
- Demicontractive operator
- Double inertial steps
- Self-adaptive algorithm
- Signal processing
- Split common fixed point problem