Skip to main content
SpringerLink
Log in

A new accelerated self-adaptive stepsize algorithm with excellent stability for split common fixed point problems

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

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

    Article  MathSciNet  Google Scholar 

  • An NT, Nam NM, Qin X (2020) Solving \(k\)-center problems involving sets based on optimization techniques. J Glob Optim 76:189–209

    Article  MathSciNet  Google Scholar 

  • Boikanyo OA (2015) A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput 265:844–853

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Elfving T (1994) A multiprojection algorithm using Bregman projections in a product space. Numer Algorithms 8:221–239

    Article  MathSciNet  Google Scholar 

  • Censor Y, Segal A (2009) The split common fixed point problem for directed operators. J Convex Anal 16:587–600

    MathSciNet  MATH  Google Scholar 

  • Censor Y, Gibali A, Reich S (2012) Algorithms for the split variational inequality problem. Numer Algorithms 59:301–323

    Article  MathSciNet  Google Scholar 

  • Chambolle A, Lions PL (1997) Image recovery via total variation minimization and related problems. Numer Math 76:167–188

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Cho SY, Kang SM (2012) Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math Sci 32:1607–1618

    Article  MathSciNet  Google Scholar 

  • Combettes PL, Hirstoaga SA (2005) Equilibrium programming in Hilbert spaces. J Nonlinear Convex Anal 6:117–136

    MathSciNet  MATH  Google Scholar 

  • Cui H, Wang F (2014) Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theory Appl 2014:1–8

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Karpagam S, Agrawal S (2011) Best proximity point theorems for cyclic orbital Meir-Keeler contraction maps. Nonlinear Anal 74:1040–1046

    Article  MathSciNet  Google Scholar 

  • Kraikaew R, Saejung S (2014) On split common fixed point problems. J Math Anal Appl 415(2):513–524

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Maingé PE, Moudafi A (2008) Convergence of new inertial proximal methods for DC programming. SIAM J Optim 19:397–413

    Article  MathSciNet  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Marino G, Xu HK (2007) Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J Math Anal Appl 329:336–346

    Article  MathSciNet  Google Scholar 

  • Meir A, Keeler E (1969) A theorem on contraction mappings. J Math Anal Appl 28:326–329

    Article  MathSciNet  Google Scholar 

  • Moudafi A (2000) Viscosity approximation methods for fixed-points problems. J Math Anal Appl 241:46–55

    Article  MathSciNet  Google Scholar 

  • Moudafi A (2010) The split common fixed-point problem for demicontractive mappings. Inverse Prob 26:055007

    Article  MathSciNet  Google Scholar 

  • Moudafi A (2011) A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal 74:4083–4087

    Article  MathSciNet  Google Scholar 

  • Nikolova M (2004) A variational approach to remove outliers and impulse noise. J Math Imaging Vis 20(1–2):99–120

    Article  MathSciNet  Google Scholar 

  • Qin X, An NT (2019) Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets. Comput Optim Appl 74:821–850

    Article  MathSciNet  Google Scholar 

  • Qin X, Yao JC (2019) A viscosity iterative method for a split feasibility problem. J Nonlinear Convex Anal 20:1497–1506

    MathSciNet  Google Scholar 

  • Shehu Y, Agbebaku DF (2018) On split inclusion problem and fixed point problem for multi-valued mappings. Comp Appl Math 37:1807–1824

    Article  MathSciNet  Google Scholar 

  • Suzuki T (2007) Moudafi’s viscosity approximations with Meir-Keeler contractions. J Math Anal Appl 325:342–352

    Article  MathSciNet  Google Scholar 

  • Takahashi W (2017) The split common fixed point problem and the shrinking projection method in Banach spaces. J Convex Anal 24:1015–1028

    MathSciNet  MATH  Google Scholar 

  • 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

    Article  MathSciNet  Google Scholar 

  • Wang F (2017) A new iterative method for the split common fixed point problem in Hilbert spaces. Optimization 66:407–415

    Article  MathSciNet  Google Scholar 

  • Yao Y, Liou YC, Postolache M (2018) Self-adaptive algorithms for the split problem of the demicontractive operators. Optimization 67:1309–1319

    Article  MathSciNet  Google Scholar 

  • Zhou HY (2008) Convergence theorems of fixed points for \(\kappa \)-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal 69:456–462

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Zheng Zhou, Bing Tan & Songxiao Li

Authors
  1. Zheng Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bing Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Songxiao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zheng Zhou.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s40314-020-01237-0

Keywords

Mathematics Subject Classification

Search

Navigation