Abstract
In the framework of Hilbert spaces, we study the solutions of split common fixed point problems. A new accelerated self-adaptive stepsize algorithm with excellent stability is proposed under the effects of inertial techniques and Meir–Keeler contraction mappings. The strong convergence theorems are obtained without prior knowledge of operator norms. Finally, in applications, our main results in this paper are applied to signal recovery problems.
Similar content being viewed by others
References
Alvarez F, Attouch H (2001) An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set Valu Anal 9:3–11
An NT, Nam NM, Qin X (2020) Solving \(k\)-center problems involving sets based on optimization techniques. J Glob Optim 76:189–209
Boikanyo OA (2015) A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput 265:844–853
Censor Y, Elfving T (1994) A multiprojection algorithm using Bregman projections in a product space. Numer Algorithms 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, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Algorithms 59:301–323
Chambolle A, Lions PL (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188
Chang SS, Wen CF, Yao JC (2018) Common zero point for a finite family of inclusion problems of accretive mappings in Banach spaces. Optimization 67:1183–1196
Cho SY, Kang SM (2012) Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci 32:1607–1618
Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117–136
Cui H, Wang F (2014) Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theory Appl 2014:1–8
He SN, Yang CP (2013) Solving the variational inequality problem defined on intersection of finite level sets. Abstr Appl Anal. https://doi.org/10.1155/2013/942315
He H, Liu S, Chen R, Wang X (2016) Strong convergence results for the split common fixed point problem. J Nonlinear Sci Appl 9:5332–5343
Karpagam S, Agrawal S (2011) Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal 74:1040–1046
Kraikaew R, Saejung S (2014) On split common fixed point problems. J Math Anal Appl 415(2):513–524
López G, Martín-Márquez V, Wang F, Xu HK (2012) Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Prob 28:085004
Maingé PE, Moudafi A (2008) Convergence of new inertial proximal methods for DC programming. SIAM J Optim 19:397–413
Majee P, Nahak C (2018) A modified iterative method for split problem of variational inclusions and fixed point problems. Comp Appl Math 37:4710–4729
Marino G, Xu HK (2007) Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 329:336–346
Meir A, Keeler E (1969) A theorem on contraction mappings. J Math Anal Appl 28:326–329
Moudafi A (2000) Viscosity approximation methods for fixed-points problems. J Math Anal Appl 241:46–55
Moudafi A (2010) The split common fixed-point problem for demicontractive mappings. Inverse Prob 26:055007
Moudafi A (2011) A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal 74:4083–4087
Nikolova M (2004) A variational approach to remove outliers and impulse noise. J Math Imaging Vis 20(1–2):99–120
Qin X, An NT (2019) Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput Optim Appl 74:821–850
Qin X, Yao JC (2019) A viscosity iterative method for a split feasibility problem. J Nonlinear Convex Anal 20:1497–1506
Shehu Y, Agbebaku DF (2018) On split inclusion problem and fixed point problem for multi-valued mappings. Comp Appl Math 37:1807–1824
Suzuki T (2007) Moudafi’s viscosity approximations with Meir-Keeler contractions. J Math Anal Appl 325:342–352
Takahashi W (2017) The split common fixed point problem and the shrinking projection method in Banach spaces. J Convex Anal 24:1015–1028
Vaish R, Ahmad MK (2020) Generalized viscosity implicit scheme with Meir-Keeler contraction for asymptotically nonexpansive mapping in Banach spaces. Numer Algorithms 84:1217–1237
Wang F (2017) A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66:407–415
Yao Y, Liou YC, Postolache M (2018) Self-adaptive algorithms for the split problem of the demicontractive operators. Optimization 67:1309–1319
Zhou HY (2008) Convergence theorems of fixed points for \(\kappa \)-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal 69:456–462
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Baisheng Yan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhou, Z., Tan, B. & Li, S. A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems. Comp. Appl. Math. 39, 220 (2020). https://doi.org/10.1007/s40314-020-01237-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-020-01237-0