Strong convergence theorem for split variational inclusion problem and finite family of fixed point problems

Document Type : Research Paper


Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh 202002, India


The main objective of this paper is to introduce and study a new type of iterative method to approximate a common solution of split variational inclusion problem and a finite family of fixed point problems in real Hilbert spaces. Furthermore, we show that the sequence generated by the proposed iterative method converges strongly to a common solution to these problems. The method and results presented in this paper extend and unify some recent known results in this field. Finally, a numerical example is used to demonstrate the convergence analysis of the sequences generated by the iterative method.


[1] M. Abdellatif, Split monotone variational inclusions, J. Optim. Theory Appl. 150 (2011), no. 2, 275–283.
[2] C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inv. Prob. 18 (2002), no. 2, 441.
[3] C. Byrne, Y. Censor, A. Gibali, and S. Reich, The split common null point problem, J. Nonlinear Convex Anal. 13 (2012), no. 4, 759–775.
[4] Y. Censor, T. Bortfeld, B. Martin, and A. Trofimov, A unified approach for inversion problems in intensitymodulated radiation therapy, Phys. Med. Bio. 51 (2006), no. 10, 2353.
[5] Y. Censor and T. Elfving, A multiprojection algorithm using bregman projections in a product space, Numerical Algorithms 8 (1994), no. 2, 221–239.
[6] Y. Censor, A. Gibali, and S. Reich, Algorithms for the split variational inequality problem, Numerical Algorithms 59 (2012), no. 2, 301–323.
[7] P.L. Combettes, The convex feasibility problem in image recovery, Adv. Imag. Electron Phys., vol. 95, Elsevier, 1996, pp. 155–270.
[8] , Solving monotone inclusions via compositions of nonexpansive averaged operators, Optim. 53 (2004), no. 5-6, 475–504.
[9] M. Dilshad, A.H. Siddiqi, R. Ahmad, and F.A. Khan, An iterative algorithm for a common solution of a split variational inclusion problem and fixed point problem for non-expansive semigroup mappings, Ind. Math. Complex Syst., Springer, 2017, pp. 221–235.
[10] K. Goebel and W.A. Kirk, Topics in metric fixed point theory, no. 28, Cambridge university press, 1990.
[11] X. Hong-Kun, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), no. 3, 659–678.
[12] K.R. Kazmi and S.H. Rizvi, An iterative method for split variational inclusion problem and fixed point problem for a nonexpansive mapping, Optimi. Lett. 8 (2014), no. 3, 1113–1124.
[13] S. Kesornprom and P. Cholamjiak, Proximal type algorithms involving linesearch and inertial technique for split variational inclusion problem in hilbert spaces with applications, Optim. 68 (2019), no. 12, 2369–2395.
[14] P. Majee and C. Nahak, A modified iterative method for split problem of variational inclusions and fixed point problems, Comput. Appl. Math. 37 (2018), no. 4, 4710–4729.
[15] G. Marino and H.-K. Xu, A general iterative method for nonexpansive mappings in hilbert spaces, J. Math. Anal. Appl. 318 (2006), no. 1, 43–52.
[16] A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inv. Prob. 26 (2010), no. 5,055007.
[17] Tomonari Suzuki, Strong convergence of krasnoselskii and mann’s type sequences for one-parameter nonexpansive semigroups without bochner integrals, J. Math. Anal. Appl. 305 (2005), no. 1, 227–239.
[18] H.-K. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150 (2011), no. 2, 360–378.
[19] H. Zhou, Convergence theorems of fixed points for κ-strict pseudo-contractions in hilbert spaces, Nonlinear Anal.: Theory Meth. Appl. 69 (2008), no. 2, 456–462.
[20] H. Zhou and P. Wang, A simpler explicit iterative algorithm for a class of variational inequalities in hilbert spaces, J. Optim. Theory Appl. 161 (2014), no. 3, 716–727.
Volume 14, Issue 1
January 2023
Pages 2425-2438
  • Receive Date: 18 November 2021
  • Revise Date: 27 October 2022
  • Accept Date: 27 December 2022