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An Inertial Type Algorithm for Extended Split Equality Variational Inclusion and Fixed Point Problems


In this article, we study an extended split equality variational inclusion and fixed point problems which is an extension of the split equality variational inclusion problems and fixed point problems. We propose a simultaneous inertial type iterative algorithm with a self adaptive stepsizes such that there is no need for a prior information about the operator norm. We further stated and prove that the proposed algorithm weakly converges to a solution of the extended split equality variational inclusion and fixed point problems. Finally, we give some numerical examples to demonstrate the performance and the applicability of the proposed algorithm. The results of this paper complements and extends results on split equality variational inclusion and fixed point problems.

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  1. 1.

    Ali, R., Kazmi, K.R., Farid, M.: Viscosity iterative method for a split equality monotone variational inclusion problem. Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms 26, 313–344 (2019)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Alvarez, F., Attouch, H.: An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9, 3–11 (2001)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Attouch, H., Bolte, J., Redont, P., Soubeyran, A.: Alternating proximal algorithms for weakly coupled minimization problems. Applications to dynamical games and PDE’s. J. Convex Anal. 28, 39–44 (2008)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bot, R.I., Csetnek, E.R., Hendrich, C.: Inertial Douglas–Rachford splitting for monotone inclusion. Appl. Math. Comput. 256, 472–487 (2015)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Bot, R.I., Csetnek, E.R.: An inertial forward–backward–forward primal-dual splitting algorithm for solving monotone inclusion problems. Numer. Algorithms 71, 519–540 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Byrne, C.: Iterative oblique projection onto convex sets and the split feasibility problem. Inverse Probl. 18, 441–453 (2002)

    MathSciNet  Article  Google Scholar 

  7. 7.

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

    MathSciNet  Article  Google Scholar 

  8. 8.

    Byrne, C., Censor, Y., Gibali, A., Reich, S.: Weak and strong convergence of algorithms for the split common null point problem. J. Nonlinear Convex Anal. 13, 759–775 (2012)

    MathSciNet  MATH  Google Scholar 

  9. 9.

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

    Article  Google Scholar 

  10. 10.

    Che, H., Chen, H., Li, M.: A new simultaneous iterative method with a parameter for solving the extended split equality fixed point problem. Numer. Algorithms 79, 1231–1256 (2018)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dong, Q.L., He, S.N., Zhao, J.: Solving the split equality problem without prior knowledge of operator norms. Optimization 64, 1887–1906 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dong, Q.L., Cho, Y.J., Zhong, L.L.: Inertial projection and contraction algorithms for variational inequalities. J. Glob. Optim. 70, 687–704 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Eslamian, M.: General algorithms for split common fixed point problem of demicontractive mappings. Optimization 65, 443–465 (2016)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Eslamian, M., Latif, A.: General split feasibility problems in Hilbert spaces. Abstr. Appl. Anal. 2013, 805104 (2013)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Eslamian, M., Vahidi, J.: Split common fixed point problem of nonexpansive semigroup. Mediterr. J. Math. 13, 1177–1195 (2016)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)

    Book  Google Scholar 

  17. 17.

    Guo, H., He, H., Chen, R.: Strong convergence theorems for the split equality variational inclusion problem and fixed point problem in Hilbert spaces. Fixed Point Theory Appl. 2015, 223 (2015)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Iyiola, O.S., Ogbuisi, F.U., Shehu, Y.: An inertial type iterative method with Armijo linesearch for nonmonotone equilibrium problems. Calcolo 55, 52 (2018)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Kazmi, K.R., Rizvi, S.H.: An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping. Optim. Lett. 8, 1113–1124 (2014)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Korpelevic, G.M.: An extragradient method for finding saddle points and for other problems (Russian). Ékonom. i Mat. Metody 12, 747–756 (1976)

    MathSciNet  Google Scholar 

  21. 21.

    Latif, A., Vahidi, J., Eslamian, M.: Strong convergence for generalized multiple-set split feasibility problem. Filomat 30, 459–467 (2016)

    MathSciNet  Article  Google Scholar 

  22. 22.

    López, G., Martín-Márquez, V., Wang, F.H., Xu, H.K.: Solving the split feasibility problem without prior knowledge of matrix norms. Inverse Probl. 28, 085004 (2012)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Maingé, P.E.: Convergence theorem for inertial KM-type algorithms. J. Comput. Appl. Math. 219, 223–236 (2008)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Maingé, P.E.: Regularized and inertial algorithms for common fixed points of nonlinear operators. J. Math. Anal. Appl. 344, 876–887 (2008)

    MathSciNet  Article  Google Scholar 

  25. 25.

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

    MathSciNet  Article  Google Scholar 

  26. 26.

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

    MathSciNet  Article  Google Scholar 

  27. 27.

    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 

  28. 28.

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

    Google Scholar 

  29. 29.

    Rockafellar, R.T.: On the maximal monotonicity of subdifferential mappings. Pac. J. Math. 33, 209–216 (1970)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Takahashi, W.: Nonlinear Functional Analysis. Fixed Point Theory and Its Application. Yokohama Publishers, Yokohama (2000)

    MATH  Google Scholar 

  31. 31.

    Thong, D.V., Hieu, D.V.: Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer. Algorithms 80, 1283–1307 (2019)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Tian, M., Tong, M.: Self-adaptive subgradient extragradient method with inertial modification for solving monotone variational inequality problems and quasi-nonexpansive fixed point problems. J. Inequal. Appl. 2019(7), 19 (2019)

    MathSciNet  Google Scholar 

  33. 33.

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

    MathSciNet  Article  Google Scholar 

  34. 34.

    Zhao, J.: Solving split equality fixed-point problem of quasi-nonexpansive mappings without prior knowledge of operators norms. Optimization 64, 2619–2630 (2014)

    MathSciNet  Article  Google Scholar 

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This work is based on the research supported wholly by the National Research Foundation (NRF) of South Africa (Grant number 111992). Opinions expressed and conclusions arrived are those of the authors and are not necessarily to be attributed to the NRF.

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Correspondence to Ferdinard Udochukwu Ogbuisi.

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Communicated by Behzad Djafari-Rouhani.

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Ogbuisi, F.U. An Inertial Type Algorithm for Extended Split Equality Variational Inclusion and Fixed Point Problems. Bull. Iran. Math. Soc. (2021).

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  • Inertial term
  • Maximal monotone mapping
  • Resolvent
  • Extended spit equality variational inclusion problem
  • Fixed point problem

Mathematics Subject Classification

  • 47H05
  • 47H06
  • 47H30
  • 47J05
  • 47J25