Based on the extragradient-like method combined with shrinking projection, we propose two algorithms, the first algorithm is obtained using sequential computation of extragradient-like method and the second algorithm is obtained using parallel computation of extragradient-like method, to find a common point of the set of fixed points of a nonexpansive mapping and the solution set of the equilibrium problem of a bifunction given as a sum of the finite number of H ̈older continuous bifunctions. The convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for the bifunction and its summands
 S. Alizadeh and F. Moradlou, A strong convergence theorem for equilibrium problems and generalized hybrid mappings, Mediterr. J. Math., 13 (2016), 379-390.
 P. N. Anh, A hybrid extragradient method extended to fixed point problems and equilibrium problems, Optimization, a62 (2013), 271-283.
 P. N. Anh and L. D. Muu, A hybrid subgradient algorithm for nonexpansive mappings and equilibrium problems, Optim. Lett., 8 (2014), 727-738.
 H. H. Bauschke and P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, New York: Springer., 408, 2011.
 L. C. Ceng, S. M. Guu, H. Y. Hu, and J. C. Yao, Hybrid shrinking projection method for a generalized equilibrium problem, a maximal monotone operator and a countable family of relatively nonexpansive mappings, Comput. Math. Appl., 61 (2011), 2468–2479.
 P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal., 6 (2005), 117-136.
 J. Contreras, M. Klusch, and J. B. Krawczyk, Numerical solutions to Nash-Cournot equilibria in coupled constraint electricity markets, IEEE Transactions on Power Systems., 19 (2004), 195-206.
 P. Daniele, F. Giannessi, and A. eds Maugeri, Equilibrium problems and variational models, Dordrecht: Kluwer Academic Publishers, 68, 2003.
 K. Fan, A minimax inequality and applications, Inequalities III (O. Shisha, ed.). Academic Press, New York; 1972.
 S. D. Fl˚am and A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1996), 29-41.
 K. Goebel and W. A. Kirk, Topics in metric fixed point theory, vol. 28. Cambridge University Press, 1990.
 T. N. Hai and N. T. Vinh, Two new splitting algorithms for equilibrium problems, Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Math. RACSAM, 111 (2017), 1051-1069.
 D. V. Hieu, A new shrinking gradient-like projection method for equilibrium problems, Optimization, 66 (2017), 2291–2307.
 C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Non- linear Anal., 11 (2006), 2400-2411.
 A. Moudafi and M. Th´era, Proximal and dynamical approaches to equilibrium problems, InIll-Posed Variational Problems and Regularization Techniques. Springer, Berlin, Heidelberg, (1999), 187-201
 A. Moudafi, A Barycentric projected-subgradient algorithm for equilibrium problems, J. Nonlinear Var. Anal., 1 (2017), 43-59.
 A. Moudafi, On the convergence of splitting proximal methods for equilibrium problems in Hilbert spaces, J. Math. Anal. Appl., 359 (2009), 508-513.
 K. Nakajo, W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
 K. A. Pham and N. H. Trinh, Splitting extragradient-like algorithms for strongly pseudomonotone equilibrium problems, Numer. Algorithms., 76 (2017), 67-91.
 S. Plubtieng, R. Punpaeng, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl., 336 (2007), 455-469.
 R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. control optim., 14 (1976), 877-898.
 P. Santos, S. Scheimberg, An inexact subgradient algorithm for equilibrium problems, Compu. Appl. Math., 30 (2011), 91-107.
 A. Tada and W. Takahashi, Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, J. Optim. Theory Appl., 133 (2007), 359-370.
 S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point prob- lems in Hilbert spaces, J. Math. Anal. Appl., 331 (2007), 506-515.
 W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl., 341 (2008), 276-286.
 D. Van Hieu, L. D. Muu, and P. K. Anh, Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings, Numer. Algorithms., 47 (2016), 197–217.
 U. Witthayarat, A. A. Abdou, and Y. J. Cho, Shrinking projection methods for solving split equilibrium problems and fixed point problems for asymptotically nonexpansive mappings in Hilbert spaces, Fixed Point Theory Appl., 2015 (2015), 1–14.
 T. Zang, Y. Xiang, and J. Yang, The tripartite game model for electricity pricing in consideration of the power quality, Energies. , 10 (2017), 20-25.
Bedane, D., Gebrie, A. (2022). Hybrid shrinking projection extragradient-like algorithms for equilibrium and fixed point problems. Computational Methods for Differential Equations, 10(3), 639-655. doi: 10.22034/cmde.2021.44502.1879
Dejene Shewakena Bedane; Anteneh Getachew Gebrie. "Hybrid shrinking projection extragradient-like algorithms for equilibrium and fixed point problems". Computational Methods for Differential Equations, 10, 3, 2022, 639-655. doi: 10.22034/cmde.2021.44502.1879
Bedane, D., Gebrie, A. (2022). 'Hybrid shrinking projection extragradient-like algorithms for equilibrium and fixed point problems', Computational Methods for Differential Equations, 10(3), pp. 639-655. doi: 10.22034/cmde.2021.44502.1879
Bedane, D., Gebrie, A. Hybrid shrinking projection extragradient-like algorithms for equilibrium and fixed point problems. Computational Methods for Differential Equations, 2022; 10(3): 639-655. doi: 10.22034/cmde.2021.44502.1879