Skip to main content
Springer Nature Link
Log in

Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, motivated by the works of Kohsaka and Takahashi (SIAM J Optim 19:824–835, 2008) and Aoyama et al. (J Nonlinear Convex Anal 10:131–147, 2009) on the class of mappings of firmly nonexpansive type, we explore some properties of firmly nonexpansive-like mappings [or mappings of type (P)] in p-uniformly convex and uniformly smooth Banach spaces. We then study the split common fixed point problems for mappings of type (P) and Bregman weak relatively nonexpansive mappings in p-uniformly convex and uniformly smooth Banach spaces. We propose an inertial-type shrinking projection algorithm for solving the two-set split common fixed point problems and prove a strong convergence theorem. Also, we apply our result to the split monotone inclusion problems and illustrate the behaviour of our algorithm with several numerical examples s. The implementation of the algorithm does not require a prior knowledge of the operator norm. Our results complement many recent results in the literature in this direction. To the best of our knowledge, it seems to be the first to use the inertial technique to solve the split common fixed point problems outside Hilbert spaces.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Agarwal, R.P., Regan, D.O., Sahu, D.R.: Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, Berlin (2009)

    Google Scholar 

  2. Alakoya, T.O., Jolaoso, L.O., Mewomo, O.T.: Modified inertia subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems. Optimization (2020). https://doi.org/10.1080/02331934.2020.1723586

    Article  MATH  Google Scholar 

  3. Alber, Y., Butnariu, D.: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl. 92(1), 33–61 (1997)

    Article  MathSciNet  Google Scholar 

  4. Ansari, Q.H., Rehan, A.: Split feasibility and fixed point problems. In: Ansari, Q.H. (ed.) Nonlinear Analysis: Approximation Theory, Optimization and Application, pp. 281–322. Springer, New York (2014)

    MATH  Google Scholar 

  5. Aoyama, K., Kohsaka, F., Takahashi, W.: Strong convergence theorems for a family of mappings of type (P) and applications. In: Nonlinear Analysis and Optimization, pp. 1–17. Yokohama Publishers, Yokohama (2009)

  6. Aoyama, K., Kohsaka, F., Takahashi, W.: Three generalizations of firmly nonexpansive mappings: their relations and continuity properties. J. Nonlinear Convex Anal. 10, 131–147 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Bauschke, H.H., Borwein, J.M., Combettes, P.L.: Bregman monotone optimization algorithms. SIAM J. Control Optim. 42(2), 596–636 (2003)

    Article  MathSciNet  Google Scholar 

  8. Bello Cruz, J.Y., Shehu, Y.: An iterative method for split inclusion problems without prior knowledge of operator norms. J. Fixed Point Theory Appl. 19, 2017–2036 (2017). https://doi.org/10.1007/s11784-016-0387-8

    Article  MathSciNet  MATH  Google Scholar 

  9. Borwein, J.M., Reich, S., Sabach, S.: A characterization of Bregman firmly nonexpansive operators using a new monotonicity concept. J. Nonlinear Convex Anal. 12(1), 161–184 (2011)

    MathSciNet  MATH  Google Scholar 

  10. Bryne, C.: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 18(2), 441–453 (2002)

    Article  Google Scholar 

  11. Byrne, C., Censor, Y., Gibali, A., Reich, S.: The split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Censor, Y., Bortfeld, T., Martin, B., Trofimov, A.: A unified approach for inversion problems in intensity-modulated radiation therapy. Phys. Med. Biol. 51, 2353–2365 (2006)

    Article  Google Scholar 

  13. Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8(2), 221–239 (1994)

    Article  MathSciNet  Google Scholar 

  14. Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Probl. 21, 2071–2084 (2005)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  16. Chen, J., Wan, Z., Yuan, L., Zheng, Y.: Approximation of fixed points of weak Bregman relatively nonexpansive mappings in Banach spaces. IJMMS (2011). Article ID 420192

  17. Chidume, C.E.: Geometric Properties of Banach Spaces and Nonlinear Iterations, Lecture Notes in Mathematics 1965, vol. 1965. Springer, London (2009)

    Book  Google Scholar 

  18. Cholamjiak, P., Thong, D.V., Cho, Y.J.: A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl. Math. (2019). https://doi.org/10.1007/s10440-019-00297-7

    Article  Google Scholar 

  19. Gibali, A., Jolaoso, L.O., Mewomo, O.T., Taiwo, A.: Fast and simple Bregman projection methods for solving variational inequalities and related problems in Banach spaces. Results Math. 75 (2020). Art. No. 179

  20. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Marcel Dekker, New York (1984)

    MATH  Google Scholar 

  21. Izuchukwu, C., Mebawondu, A.A., Mewomo, O.T.: A new method for solving split variational inequality problems without co-coerciveness. J. Fixed Point Theory Appl. 22(4), 1–23 (2020)

    Article  MathSciNet  Google Scholar 

  22. Izuchukwu, C., Ogwo, G.N., Mewomo, O.T.: An inertial method for solving generalized split feasibility problems over the solution set of monotone variational inclusions. Optimization (2020). https://doi.org/10.1080/02331934.2020.1808648

    Article  Google Scholar 

  23. Izuchukwu, C., Ugwunnadi, G.C., Mewomo, O.T., Khan, A.R., Abbas, M.: Proximal-type algorithms for split minimization problem in p-uniformly convex metric space. Numer. Algorithms 82(3), 909–935 (2019)

    Article  MathSciNet  Google Scholar 

  24. Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: A parallel combination extragradient method with Armijo line searching for finding common solution of finite families of equilibrium and fixed point problems. Rend. Circ. Mat. Palermo II 69(3), 711–735 (2020)

    Article  MathSciNet  Google Scholar 

  25. Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: Strong convergence theorem for solving pseudo-monotone variational inequality problem using projection method in a reflexive Banach space. J. Optim. Theory Appl. 185(3), 744–766 (2020)

    Article  MathSciNet  Google Scholar 

  26. Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A strong convergence theorem for solving variational inequalities using an inertial viscosity subgradient extragradient algorithm with self adaptive stepsize. Demonstr. Math. 52(1), 183–203 (2019)

    Article  MathSciNet  Google Scholar 

  27. Jolaoso, L.O., Taiwo, A., Alakoya, T.O., Mewomo, O.T.: A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem. Comput. Appl. Math. 39(1), 38 (2020)

    Article  MathSciNet  Google Scholar 

  28. Jolaoso, L.O., Alakoya, T.O., Taiwo, A., Mewomo, O.T.: Inertial extragradient method via viscosity approximation approach for solving Equilibrium problem in Hilbert space. Optimization (2020). https://doi.org/10.1080/02331934.2020.1716752

    Article  MATH  Google Scholar 

  29. Kohsaka, F., Takahashi, W.: Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces. SIAM J. Optim. 19, 824–835 (2008)

    Article  MathSciNet  Google Scholar 

  30. Lindenstrauss, J., Tzafriri, L.: Classical Banach Spaces, vol. II. Springer, Berlin (1979)

    Book  Google Scholar 

  31. Lin, L.J., Chen, Y.D., Chuang, C.S.: Solutions for a variational inclusion problem with applications to multiple sets split feasibility problems. Fixed Point Theory Appl. 2013, 333 (2013)

    Article  MathSciNet  Google Scholar 

  32. Moudafi, A.: A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 74(12), 4083–4087 (2011)

    Article  MathSciNet  Google Scholar 

  33. Moudafi, A.: Split monotone variational inclusions. J. Optim. Theory Appl. 150, 275–283 (2011)

    Article  MathSciNet  Google Scholar 

  34. Naraghirad, E., Yao, J.C.: Bregman weak relatively non expansive mappings in Banach space. Fixed Point Theory Appl. (2013). https://doi.org/10.1186/1687-1812-2013-141

    Article  MATH  Google Scholar 

  35. Ogwo, G.N., Izuchukwu, C., Aremu, K.O., Mewomo, O.T.: A viscosity iterative algorithm for a family of monotone inclusion problems in an Hadamard space. Bull. Belg. Math. Soc. Simon Stevin 27(1), 127–152 (2020)

    Article  MathSciNet  Google Scholar 

  36. Phelps, R.R.: Convex Functions, Monotone Operators, and Differentiability. Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin (1993)

    Google Scholar 

  37. Polyak, B.T.: Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys. 4, 1–17 (1964)

    Article  Google Scholar 

  38. Reich, S., Sabach, S.: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, pp. 299-314. Springer, New York (2010)

  39. Sch\(\ddot{o}\)pfer, F.: Iterative regularization method for the solution of the split feasibility problem in Banach spaces. Ph.D thesis, Saabr\(\ddot{u}\)cken (2007)

  40. Schopfer, F., Schuster, T., Louis, A.K.: An iterative regularization method for the solution of the split feasibility problem in Banach spaces. Inverse Probl. 24(5), 55008 (2008)

    Article  MathSciNet  Google Scholar 

  41. Shehu, Y., Mewomo, O.T.: Further investigation into split common fixed point problem for demicontractive operators. Acta Math. Sin. (Engl. Ser.) 32, 1357–1376 (2016)

    Article  MathSciNet  Google Scholar 

  42. Shehu, Y., Vuong, P.T., Cholamjiak, P.: A self-adaptive projection method with an inertial technique for split feasibility problems in Banach spaces with applications to image restorations problems. J. Fixed Point Theory Appl. 21, 50 (2019). https://doi.org/10.1007/s11784-019-0684-0

    Article  MathSciNet  MATH  Google Scholar 

  43. Song, Y.: Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces. J. Nonlinear Sci. Appl. 11, 198–217 (2018)

    Article  MathSciNet  Google Scholar 

  44. Su, Y., Wang, Z., Xu, H.: Strong convergence theorems for a common fixed point of two hemi-relatively nonexpansive mappings. Nonlinear Anal. 71, 5616–5628 (2009)

    Article  MathSciNet  Google Scholar 

  45. Suparatulatorn, R., Khemphet, A., Charoensawan, P., Suantai, S., Phudolsitthiphat, N.: Generalized self-adaptive algorithm for solving split common fixed point problem and its application to image restoration problem. Int. J. Comput. Math. 97(7), 1431–1443 (2020). https://doi.org/10.1080/00207160.2019.1622687

    Article  MathSciNet  Google Scholar 

  46. Taiwo, A., Alakoya, T.O., Mewomo, O.T.: Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces. Numer. Algorithms (2020). https://doi.org/10.1007/s11075-020-00937-2

    Article  Google Scholar 

  47. Taiwo, A., Jolaoso, L. O., Mewomo, O. T.: A modified Halpern algorithm for approximating a common solution of split equality convex minimization problem and fixed point problem in uniformly convex Banach spaces. Comput. Appl. Math. 38(2) (2019). Article 77

  48. Taiwo, A., Jolaoso, L.O., Mewomo, O.T.: Parallel hybrid algorithm for solving pseudomonotone equilibrium and Split Common Fixed point problems. Bull. Malays. Math. Sci. Soc. 43(2), 1893–1918 (2019)

    Article  MathSciNet  Google Scholar 

  49. Taiwo, A., Jolaoso, L.O., Mewomo, O.T., Gibali, A.: On generalized mixed equilibrium problem with \(\alpha \)-\(\beta \)-\(\mu \) bifunction and \(\mu \)-\(\tau \) monotone mapping. J. Nonlinear Convex Anal. 21(6), 1381–1401 (2020)

    MathSciNet  Google Scholar 

  50. Taiwo, A., Owolabi, A.O.-E., Jolaoso, L.O., Mewomo, O.T., Gibali, A.: A new approximation scheme for solving various split inverse problems. Afr. Mat. (2020). https://doi.org/10.1007/s13370-020-00832-y

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  52. Takahashi, W.: The split common fixed point problem for generalized demimetric mappings in two Banach spaces. Optimization (2018). https://doi.org/10.1080/02331934.2018.1522637

    Article  MATH  Google Scholar 

  53. Tang, J., Chang, S., Wang, L., Wang, X.: On the split common fixed point problem for strict pseudocontractive and asymptotically nonexpansive mappings in Banach spaces. J. Inequalities Appl. 2015, 305 (2015)

    Article  MathSciNet  Google Scholar 

  54. Thong, D.V., Hieu, D.V.: An inertial method for solving split common fixed point problems. J. Fixed Point Theory Appl. (2017). https://doi.org/10.1007/s11784-017-0464-7

    Article  MathSciNet  MATH  Google Scholar 

  55. Tuyen, T.M., Thuy, N.T.T., Trang, N.M.: A strong convergence theorem for a parallel iterative method for solving the split common null point problem in Hilbert spaces. J. Optim. Theory Appl. (2019). https://doi.org/10.1007/s10957-019-01523-w

    Article  MathSciNet  MATH  Google Scholar 

  56. Xu, Z.B., Roach, G.F.: Characteristic inequalities of uniformly convex and uniformly smooth Banach spaces. J. Math. Anal. Appl. 157(1), 189–210 (1991)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors sincerely thank the anonymous reviewer for his careful reading, constructive comments and fruitful suggestions that substantially improved the manuscript. The first author acknowledges with thanks the International Mathematical Union Breakout Graduate Fellowship (IMU-BGF) Award for his doctoral study. The second author acknowledges with thanks the bursary and financial support from Department of Science and Technology and National Research Foundation, Republic of South Africa Center of Excellence in Mathematical and Statistical Sciences (DST-NRF COE-MaSS) Doctoral Bursary. The third author is supported by the National Research Foundation (NRF) of South Africa Incentive Funding for Rated Researchers (Grant Number 119903). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the CoE-MaSS, IMU and NRF

Author information

Authors and Affiliations

  1. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa

    A. Taiwo, L. O. Jolaoso & O. T. Mewomo

  2. DST-NRF Center of Excellence in Mathematical and Statistical Sciences (CoE-MaSS), Johannesburg, South Africa

    L. O. Jolaoso

Authors
  1. A. Taiwo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. L. O. Jolaoso
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. O. T. Mewomo
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to O. T. Mewomo.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interests.

Additional information

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

Taiwo, A., Jolaoso, L.O. & Mewomo, O.T. Inertial-Type Algorithm for Solving Split Common Fixed Point Problems in Banach Spaces. J Sci Comput 86, 12 (2021). https://doi.org/10.1007/s10915-020-01385-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10915-020-01385-9

Keywords

Mathematics Subject Classification