Skip to main content

Advertisement

SpringerLink
Log in

A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem

  • Published:
Calcolo Aims and scope Submit manuscript

Abstract

The purpose of this paper is to introduce and study an iterative algorithm for solving a general split equality problem. The problem consists of finding a common element of the set of common zero points for a finite family of maximal monotone operators, the set of common fixed points for a finite family of demimetric mappings and the set of common solutions of variational inequality problems for a finite family of inverse strongly monotone mappings in the setting of infinite-dimensional Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence theorem for approximating a solution of the split equality problem. As special cases, we shall utilize our results to study the split equality equilibrium problems and the split equality optimization problems. Our result complements and extends some related results in literature.

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

Similar content being viewed by others

References

  1. Moudafi, A.: A relaxed alternating CQ-algorithms for convex feasibility problems. Nonlinear Anal. TMA 79, 117–121 (2013)

    Article  MathSciNet  Google Scholar 

  2. Censor, Y., Elfving, T.: A multiprojection algorithms using Bragman projection in a product space. Numer. Algorithm 8, 221–239 (1994)

    Article  Google Scholar 

  3. Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20, 103–120 (2004)

    Article  MathSciNet  Google Scholar 

  4. 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 

  5. Attouch, H., Cabot, A., Frankel, F., Peypouquet, J.: Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDEs. Nonlinear Anal. 74, 7455–7473 (2011)

    Article  MathSciNet  Google Scholar 

  6. Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDEs. J. Convex Anal. 15, 485–506 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Moudafi, A.: Alternating CQ-algorithm for convex feasibility and split fixed-point problems. J. Nonlinear Convex Anal. 15, 809–818 (2014)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  10. Masad, E., Reich, S.: A note on the multiple-set split convex feasibility problem in Hilbert space. J. Nonlinear Convex Anal. 8, 367–371 (2007)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  12. Xu, H.K.: Iterative methods for split feasibility problem in infinite-dimensional Hilbert spaces. Inverse Probl. 26, 105018 (2010)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. Chang, S.S., Agarwal, R.P.: Strong convergence theorems of general split equality problems for quasi-nonexpansive mappings. J. Inequal. Appl. 2014, 367 (2014)

    Article  MathSciNet  Google Scholar 

  15. Takahashi, W., Xu, H.K., Yao, J.C.: Iterative methods for generalized split feasibility problems in Hilbert spaces. Set-Valued Var. Anal. 23, 205–221 (2015)

    Article  MathSciNet  Google Scholar 

  16. Eslamian, M., Eslamian, P.: Strong convergence of a split common fixed point problem. Numer. Func. Anal. Optim. 37, 1248–1266 (2016)

    Article  MathSciNet  Google Scholar 

  17. Eslamian, M.: Split common fixed point and common null point problem. Math. Methods Appl. Sci. 40(18), 7410–7424 (2017)

    Article  MathSciNet  Google Scholar 

  18. Chang, S.-S., Quan, J., Liu, J.: Feasible iterative algorithms and strong convergence theorems for bi-level fixed point problems. J. Nonlinear Sci. Appl. 9, 1515–1528 (2016)

    Article  MathSciNet  Google Scholar 

  19. Xiao, Y., Censor, Y., Michalski, D., Galvin, J.M.: The least-intensity feasible solution for aperture-based inverse planning in radiation therapy. Ann. Oper. Res. 119, 183–203 (2003)

    Article  Google Scholar 

  20. Moudafi, A., Al-Shemas, E.: Simultaneous iterative methods for split equality problems and application. Trans. Math. Program. Appl. 1, 1–11 (2013)

    Google Scholar 

  21. 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 

  22. Lopez, G., Martin-Marquez, V., Wang, F., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 27, 085004 (2012)

    Article  MathSciNet  Google Scholar 

  23. Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64, 2619–2630 (2015)

    Article  MathSciNet  Google Scholar 

  24. Gibali, A.: A new split inverse problem and application to least intensity feasible solutions. Pure Appl. Funct. Anal. 2, 243–258 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Takahashi, W.: The split common fixed point problem and strong convergence theorems by hybrid methods in two Banach spaces. J. Nonlinear Convex Anal. 17, 1051–1067 (2016)

    MathSciNet  MATH  Google Scholar 

  26. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)

    Article  MathSciNet  Google Scholar 

  27. Kocourek, P., Takahashi, W., Yao, J.C.: Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces. Taiwan J. Math. 14, 2497–2511 (2010)

    Article  MathSciNet  Google Scholar 

  28. Marino, G., Xu, H.K.: Weak and strong convergence theorems for strictly pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–349 (2007)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  30. Reich, S.: Extension problems for accretive sets in Banach spaces. J. Funct. Anal. 26, 378–395 (1977)

    Article  MathSciNet  Google Scholar 

  31. Brezis, H.: Operateurs Maximaux Monotones et Semi-Groups de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam (1973)

    MATH  Google Scholar 

  32. Mainge, P.E.: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 16, 899–912 (2008)

    Article  MathSciNet  Google Scholar 

  33. Xu, H.K.: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc. 66, 240–256 (2002)

    Article  MathSciNet  Google Scholar 

  34. Zegeye, H., Shahzad, N.: Convergence of Mann’s type iteration method for generalized asymptotically nonexpansive mappings. Comput. Math. Appl. 62, 4007–4014 (2011)

    Article  MathSciNet  Google Scholar 

  35. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  36. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  37. Flam, S.D., Antipin, A.S.: Equilibrium programming using proximal-link algolithms. Math. Program. 78, 29–41 (1997)

    Article  Google Scholar 

  38. Tada, A., Takahashi, W.: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 133, 359–370 (2007)

    Article  MathSciNet  Google Scholar 

  39. Takahashi, S., Takahashi, W.: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 331, 506–515 (2007)

    Article  MathSciNet  Google Scholar 

  40. Reich, S., Sabach, S.: Three strong convergence theorems regarding iterative methods for solving equilibrium problems in reflexive Banach spaces. Contemp. Math. 568, 225–240 (2012)

    Article  MathSciNet  Google Scholar 

  41. Ceng, L.C., Petrusel, A., Yao, J.C.: Composite viscosity approximation methods for equilibrium problem, variational inequality and common fixed points. J. Nonlinear Convex Anal. 15, 219–240 (2014)

    MathSciNet  MATH  Google Scholar 

  42. Eslamian, M.: Strong convergence of a new multi-step algorithm for strict pseudo- contractive mappings and Ky Fan inequality. Mediterran. J. Math. 12, 1161–1176 (2015)

    Article  MathSciNet  Google Scholar 

  43. Iiduka, H., Yamada, I.: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 19, 1881–1893 (2009)

    Article  MathSciNet  Google Scholar 

  44. Iiduka, H., Yamada, I.: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 58, 251–261 (2009)

    Article  MathSciNet  Google Scholar 

  45. Mainge, P.E.: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 47, 1499–1515 (2008)

    Article  MathSciNet  Google Scholar 

  46. Takahashi, S., Takahashi, W., Toyoda, M.: Strong convergence theorems for maximal monotone operators with nonlinear mappings in Hilbert spaces. J. Optim. Theory Appl. 147, 27–41 (2010)

    Article  MathSciNet  Google Scholar 

  47. Aoyama, K., Kimura, Y., Takahashi, W.: Maximal monotone operators and maximal monotone functions for equilibrium problems. J. Convex Anal. 15, 395–409 (2008)

    MathSciNet  MATH  Google Scholar 

  48. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin (2012)

    MATH  Google Scholar 

  49. Zhao, J., Jia, Y., Zhang, H.: General alternative regularization methods for split equality common fixed-point problem. Optimization 67, 619–635 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of Mazandaran, P.O. Box 48518-78413, Behshahr, Iran

    Mohammad Eslamian

  2. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

  3. Department of Mathematics, Computer Science and Information Systems, California University of Pennsylvania, California, PA, USA

    Olaniyi S. Iyiola

Authors
  1. Mohammad Eslamian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yekini Shehu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Olaniyi S. Iyiola
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mohammad Eslamian.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eslamian, M., Shehu, Y. & Iyiola, O.S. A strong convergence theorem for a general split equality problem with applications to optimization and equilibrium problem. Calcolo 55, 48 (2018). https://doi.org/10.1007/s10092-018-0290-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10092-018-0290-3

Keywords

Mathematics Subject Classification

Search

Navigation