Solutions of system of split mixed equilibrium and fixed points problems

Document Type : Research Paper


1 Department of Mathematics, Faculty of Science, Erzurum Technical University, Erzurum, 25700, Türkiye

2 Department of Mathematics, University of Eswatini, Private Bag, Kwaluseni, Eswatini


In this paper, we introduce a new iterative method for a system of split mixed equilibrium problems and an infinite family of demimetric mappings in a real Hilbert space. Then, we establish that the sequence generated by our proposed algorithm converges strongly to a common element in the solutions set of a system of split mixed equilibrium problems and the common fixed points set of an infinite family of demimetric mappings. Our results improve and generalize some well-known recent results in the literature.


[1] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145.
[2] F.E. Browder and W.V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J. Math. Anal. Appl. 20 (1967), 197–228.
[3] C.E. Chidume and S. Maruster, Iterative methods for the computation of fixed points of demicontractive mappings, J. Comput. Appl. Math. 234 (2010), 861–882.
[4] P. E. Mainge, Approximation methods for common fixed points of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 325 (2007), 469–479.
[5] S. Akashi and W. Takahashi, Weak convergence theorem for an infinite family of demimetric mappings in a Hilbert space, J. Nonlinear Convex Anal. 10 (2016), 2159–2169.
[6] Z. He, The split equilibrium problem and its convergence algorithms, J. Inequal Appl. 2012 (2012), 162.
[7] D.V. Hieua, Parallel extragradient-proximal methods for split equilibrium problems, Math. Modell. Anal. 21 (2016), no. 4, 478–501.
[8] K.R. Kazmi and S.H. Rizvi, Iterative approximation of a common solution of a split equilibrium problem, a variational inequality problem and a fixed point problem, J. Egypt. Math. Soc. 21 (2013), no. 1, 44–51.
[9] A. Moudafi, The split common fixed-point problem for demicontractive mappings, Inverse Prob. 26 (2010), no. 5, Article ID 055007.
[10] N. Onjai-uea and W. Phuengrattana, On solving split mixed equilibrium problems and a fixed point problems of hybrid-type multivalued mappings in Hilbert spaces, J. Inequal. Appl. 2017 (2017), 137.
[11] M.O. Osilike and F.O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Anal. 74 (2011), 1814–1822.
[12] H. Piri, Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings, Nonlinear Anal. 74 (2011), 6788–6804.
[13] M. Rahaman, YC. Liou, R. Ahmad, and I. Ahmad, Convergence theorems for split equality generalized mixed equilibrium problems for demi-contractive mappings, J. Inequal. Appl. 2015 (2015), 418.
[14] A. Razani, Double sequence iteration for a strongly contractive mapping in the modular space, Iran. J. Math. Sci. Inform. 11 (2016), no. 2, 119–130.
[15] A. Razani and M. Bagherboum, A modified Mann iterative scheme for a sequence on nonexpansive mappings and a monotone mapping with applications, Bull. Iran. Math. Soc. 40 (2014), no. 4, 823–849.
[16] A. Razani and Z. Goodarzi, Iteration by Cesaro means for quasi-contractive mappings, Filomat 28 (2014), no. 8, 1575–1584
[17] S.H. Rizvi, A strong convergence theorem for split mixed equilibrium and fixed point problems for nonexpansive mappings, J. Fixed Point Theory Appl. (2018) 20:8.
[18] M. Shahsavari, A. Razani, and Gh. Abbasi, Some random iteration processes in modular function space, Ital. J. Pure Appl. Math. 47 (2022), 929–949.
[19] Y. Shehu and F.U. Ogbuisi, An iterative algorithm for approximating a solution of split common fixed point problem for demi-cotractive maps, Dyn. Contin. Discrete Impuls. Syst. B: Appl. Algorithms 23 (2016), 205–216.
[20] Y. Song, Iterative methods for fixed point problems and generalized split feasibility problems in Banach spaces, J. Nonlinear Sci. Appl. 11 (2018), 198–217.
[21] T. Suzuki, Strong convergence of Krasnoselskii and Mannˆas type sequences for oneparameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), no. 1, 227–239.
[22] W. Takahashi, The split common fixed point problem and the shrinking projection method in Banach spaces, J. Convex Anal. 24 (2017), no. 3, 1015–1028.
[23] W. Takahashi and K. Zembayashi, Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Anal. 70 (2009), no. 1, 45–57.
[24] G.C. Ugwunnadi and B. Ali, Approximation methods for solutions of system of split equilibrium problems, Adv. Oper. Theory 1 (2016), no. 2, 164–183.
[25] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 66 (2002), 240–256.
Volume 14, Issue 9
September 2023
Pages 1-15
  • Receive Date: 08 August 2022
  • Revise Date: 12 May 2023
  • Accept Date: 11 June 2023