math. sci. appl. e-notes Mathematical Sciences and Applications E-Notes 2147-6268 Murat TOSUN 10.36753/mathenot.1799797 Approximation Theory and Asymptotic Methods Yaklaşım Teorisi ve Asimptotik Yöntemler Solutions of Equilibrium Problems with Fixed Point Iteration Method and Data Dependence https://orcid.org/0000-0002-2083-899X Maldar Samet AKSARAY UNIVERSITY https://orcid.org/0009-0005-7630-2640 Çetin Esra MINISTRY OF NATIONAL EDUCATION OF REPUBLIC OF TÜRKİYE 02 09 2026 14 1 1 20 10 08 2025 12 31 2025 Copyright © 2013, Mathematical Sciences and Applications E-Notes 2013 Mathematical Sciences and Applications E-Notes

In this paper, we give comprehensive details about iteration methods to solve equilibrium problems through viscosity type approximation. This study introduces a new viscosity type iteration method in order to find the common point of the set of fixed points of nonexpansive mappings and relaxed $(a,b)$-cocoercive mapping, and the groups of solutions for of variational inequality and an equilibrium problem in Hilbert spaces. We also argue that this iteration method strongly converges to a common point in the mentioned three sets. As part of the applications, we give an example supporting the convergence theorem. As we emphasize the concept of data dependency for fixed point theory, we present the data dependence result for the fixed points of the stated class of mappings for the novel iteration method via viscosity type approximation.

 Equilibrium problem data dependence iteration method viscosity approximation methods [1] D. V. Thong, S. Reich, X. H. Li, P. T. H. Tham, An efficient algorithm with double inertial steps for solving split common fixed point problems and an application to signal processing, Comput. Appl. Math., 44(3) (2025), Article ID 102. https://doi.org/10.1007/s40314-024-03058-x [2] A. Büyükkaya, M. Öztürk, On Suzuki-Proinov type contractions in modular $b$-metric spaces with an application, Commun. Adv. Math. Sci., 7(1) (2024), 27-41. https://doi.org/10.33434/cams.1414411 [3] N. Adhikari,W. Sintunavarat, A novel investigation of quaternion Julia and Mandelbrot sets using the viscosity iterative approach, Results Control Optim., 18 (2025), Article ID 100525. https://doi.org/10.1016/j.rico.2025.100525 [4] M. Knefati, V. Karakaya, Iterative approximation for mean nonexpansive mappings in uniformly convex spaces with a fractional Volterra application, Univ. J. Math. Appl., 8(4) (2025), 199-210. https://doi.org/10.32323/ujma.1784049 [5] E. Hacıoğlu, F. Gürsoy, S. Maldar, Y. Atalan, G. V. Milovanovi´c, Iterative approximation of fixed points and applications to two-point second-order boundary value problems and to machine learning, Appl. Numer. Math., 167 (2021), 143-172. https://doi.org/10.1016/j.apnum.2021.04.020 [6] Y. Atalan, On a new fixed point iterative algorithm for general variational inequalities, J. Nonlinear Convex Anal., 20(11) (2019), 2371-2386. [7] I. Karahan, Strong and weak convergence theorems for split equality generalized mixed equilibrium problem, Fixed Point Theory Appl., 2016 (2016), Article ID 101. https://doi.org/10.1186/s13663-016-0592-6 [8] S. Temir, O. Korkut, Approximating fixed points of generalized $\alpha$ nonexpansive mappings by the new iteration process, J. Math. Sci. Model., 5(1) (2022), 35-39. https://doi.org/10.33187/jmsm.993823 [9] L. Muu, W. Oettli, Convergence of an adaptive penalty scheme for finding constrained equilibria, Nonlinear Anal. Theory Methods Appl., 18(12) (1992), 1159-1166. https://doi.org/10.1016/0362-546X(92)90159-C [10] E. Blum, S. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student., 63 (1994), 123–145. [11] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces , J. Math. Anal. Appl., 331(1) (2007), 506-515. https://doi.org/10.1016/j.jmaa.2006.08.036 [12] X. Qin, M. Shang, Y. Su, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Nonlinear Anal. Theory Methods Appl., 69(11) (2008), 3897-3909. https://doi.org/10.1016/j.na.2007.10.025 [13] S. S. Chang, Y. J. Cho, J. K. Kim, Approximation methods of solutions for equilibrium problem in Hilbert spaces, Dyn. Syst. Appl., 17(3-4) (2008), 503-513. [14] L. C. Ceng, N. C.Wong, Viscosity approximation methods for equilibrium problems and fixed point problems of nonlinear semigroups, Taiwan. J. Math., 13(5) (2009), 1497-1513. [15] S. S. Chang, H. J. Lee, C. K. Chan, A new method for solving equilibrium problem fixed point problem and variational inequality problem with application to optimization, Nonlinear Anal. Theory Methods Appl., 70(9) (2009), 3307-3319. https://doi.org/10.1016/j.na.2008.04.035 [16] X. Qin, S. S. Chang, Y. J. Cho, Iterative methods for generalized equilibrium problems and fixed point problems with applications, Nonlinear Anal. Real World Appl., 11(4) (2010), 2963-2972. https://doi.org/10.1016/j.nonrwa.2009.10.017 [17] P. Katchang, P. Kumam, A new iterative algorithm of solution for equilibrium problems, variational inequalities and fixed point problems in a Hilbert space, J. Appl. Math. Comput., 32(1) (2010), 19-38. https://doi.org/10.1007/s12190-009-0230-0 [18] H. Piri, A general iterative method for finding common solutions of system of equilibrium problems, system of variational inequalities and fixed point problems, Math. Comput. Model., 55(3-4) (2012), 1622-1638. https://doi.org/10.1016/j.mcm.2011.10.069 [19] S. Y. Cho, X. Qin, On the strong convergence of an iterative process for asymptotically strict pseudocontractions and equilibrium problems, Appl. Math. Comput., 235 (2014), 430-438. https://doi.org/10.1016/j.amc.2014.03.013 [20] J. Yu, N. F. Wang, Z. Yang, Equivalence results between Nash equilibrium theorem and some fixed point theorems, Fixed Point Theory Appl., 2016 (2016), Article ID 69. https://doi.org/10.1186/s13663-016-0562-z [21] D. Van Hieu, S. Reich, P. Kim Quy, Viscosity-regularization iterative methods for solving equilibrium problems in Hilbert space, Optim., 73(9) (2024), 2791-2818. https://doi.org/10.1080/02331934.2023.2231214 [22] P. V. Ndlovu, L. O. Jolaoso, M. Aphane, H. A. Abass, Viscosity extragradient with modified inertial method for solving equilibrium problems and fixed point problem in Hadamard manifold, J. Inequal. Appl., 2024(1) (2024), Article ID 21. https://doi.org/10.1186/s13660-024-03099-0 [23] R. U. Verma, Generalized system for relaxed cocoercive variational inequalities and projection methods, J. Optim. Theory Appl., 121(1) (2004), 203-210. https://doi.org/10.1023/B:JOTA.0000026271.19947.05 [24] G. Stampacchia, Formes bilinearies coercitives sur les ensembles convexes, CR Acad. Sci., 258 (1964), 4413-4416. [25] H. Iiduka,W. Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. Theory Methods Appl., 61(3), (2005) 341–350. https://doi.org/10.1016/j.na.2003.07.023 [26] R. T. Rockafellar, On the maximality of sums of nonlinear monotone operators, Trans. Am. Math. Soc., 149 (1970), 75–88. https://doi.org/10.2307/1995660 [27] S. D. Flam, A. S. Antipin, Equilibrium programming using proximal-like algorithms, Math. Program., 78 (1997), 29–41. https://doi.org/10.1007/BF02614504 [28] T. Suzuki, Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces, Fixed Point Theory Appl., 2005 (2005), Article ID 685918. https://doi.org/10.1155/FPTA.2005.103 [29] S. M. Soltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive-like operators, Fixed Point Theory Appl., 2008 (2008), Article ID 242916. https://doi.org/10.1155/2008/242916 [30] H. K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal., 298(1) (2004), 279-291. https://doi.org/10.1016/j.jmaa.2004.04.059 [31] S. Maldar, Iterative algorithms of generalized nonexpansive mappings and monotone operators with application to convex minimization problem, J. Appl. Math. Comput., 68 (2022), 1841–1868. https://doi.org/10.1007/s12190-021-01593-y