Korean J. Math. Vol. 34 No. 3 (2026) pp.409-422
DOI: https://doi.org/10.11568/kjm.2026.34.3.409

A double inertial extragradient method for finite families of mappings and its application to variational inequalities

Article Sidebar

PDF
Published: Sep 30, 2026
Keywords:
Variational Inequality Problem, Quasi-nonexpansive mapping, Fixed Point Problem
Mathematics Subject Classification:
47H09, 47H10

Main Article Content

Prashant Patel
Department of Mathematics, School of Advanced Sciences, VIT-AP University Inavolu, Beside AP Secretariat, Amaravati, 522237, Andhra Pradesh, India
https://orcid.org/0000-0003-0438-4366
Rahul Shukla
Department of Mathematical Sciences & Computing, Mthatha 5117, Walter Sisulu University, South Africa

Abstract

This paper investigates the problem of finding a common solution to a variational inequality problem involving a pseudo-monotone operator and a fixed point problem for a finite family of quasi-nonexpansive mappings in real Hilbert spaces. We introduce a new double inertial Mann-type extragradient algorithm that incorporates a non-monotonic self-adaptive step size, eliminating the need for prior knowledge of the operator's Lipschitz constant. A weak convergence theorem is established for the proposed iterative method under standard assumptions. Our work extends and generalizes recent results in the literature, particularly those of Singh and Chandok (2024), to the more general setting of a finite family of mappings.


Article Details

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported License.

References

[1] F. Alvarez and H. Attouch, An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping, Set-Valued Anal. 9 (1-2) (2001), 3–11. https://doi.org/10.1023/A:1011253113155 Google Scholar

[2] Y. Censor, A. Gibali, and S. Reich, The subgradient extragradient method for solving variational inequalities in Hilbert space, J. Optim. Theory Appl. 148 (2) (2011), 318–335. https://doi.org/10.1007/s10957-010-9757-3 Google Scholar

[3] M. Dilshad, F. M. Alamrani, A. Alamer, E. Alshaban, and M. G. Alshehri, Viscosity-type inertial iterative methods for variational inclusion and fixed point problems, AIMS Math. 9 (7) (2024), 18553–18573. https://doi.org/10.3934/math.2024903 Google Scholar

[4] F. Facchinei and J.-S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer-Verlag, New York (2003). Google Scholar

[5] D. Filali, M. Dilshad, M. Akram, M. F. Khan, and S. S. Irfan, Hybrid inertial viscosity-type forward-backward splitting algorithms for variational inclusion problems, AIMS Math. 10 (12) (2025), 28829–28860. https://doi.org/10.3934/math.20251269 Google Scholar

[6] D. Filali, M. Dilshad, A. F. Y. Alfaifi, and M. Akram, Hybrid inertial self-adaptive iterative methods for split variational inclusion problems, Axioms 14 (5) (2025), 373. https://doi.org/10.3390/axioms14050373 Google Scholar

[7] P. T. Harker, A variational inequality approach for the determination of oligopolistic market equilibrium, Math. Programming 30 (1) (1984), 105–111. https://doi.org/10.1007/BF02591802 Google Scholar

[8] G. M. Korpelevich, The extragradient method for finding saddle points and other problems, Matekon 12 (1976), 747–756. Google Scholar

[9] P. E. Maingé, Convergence theorems for inertial KM-type algorithms, J. Comput. Appl. Math. 219 (1) (2008), 223–236. https://doi.org/10.1016/j.cam.2007.07.021 Google Scholar

[10] F. O. Nwawuru, O. K. Narain, M. Dilshad, and J. N. Ezeora, Splitting method involving two-step inertial for solving inclusion and fixed point problems with applications, Fixed Point Theory Algorithms Sci. Eng. 2025 (2025), Paper No. 8, 28 pp. https://doi.org/10.1186/s13663-025-00781-w Google Scholar

[11] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (4) (1967), 591–597. https://doi.org/10.1090/S0002-9904-1967-11761-0 Google Scholar

[12] P. Patel and R. Shukla, Solving fixed point problems and variational inclusions using viscosity approximations, Aust. J. Math. Anal. Appl. 21 (2) (2024), Art. 8, 13 pp. Google Scholar

[13] P. Patel and R. Shukla, Viscosity approximation for split equality generalized mixed equilibrium problems with semigroups of nonexpansive mappings, Abstr. Appl. Anal. 2024 (2024), Art. ID 2812752, 13 pp. https://doi.org/10.1155/aaa/2812752 Google Scholar

[14] P. Patel and R. Shukla, Viscosity approximation methods for generalized modification of the system of equilibrium problem and fixed point problems of an infinite family of nonexpansive mappings, Math. Stat. 12 (4) (2024), 339–347. https://doi.org/10.13189/ms.2024.120405 Google Scholar

[15] P. Patel and R. Shukla, Hierarchical fixed point results for a countable family of strict pseudo contractive mappings in Hadamard manifolds, European J. Pure Appl. Math. 18 (2) (2025), 5951. https://doi.org/10.29020/nybg.ejpam.v18i2.5951 Google Scholar

[16] B. T. Polyak, Some methods of speeding up the convergence of iteration methods, USSR Comput. Math. Math. Phys. 4 (5) (1964), 1–17. Google Scholar

[17] Y. Shehu, O. S. Iyiola, and S. Reich, A modified inertial subgradient extragradient method for solving variational inequalities, Optim. Eng. 23 (1) (2022), 421–449. https://doi.org/10.1007/s11081-020-09593-w Google Scholar

[18] W. Singh and S. Chandok, Mann-type extragradient algorithm for solving variational inequality and fixed point problems, Comput. Appl. Math. 43 (4) (2024), Paper No. 259, 25 pp. https://doi.org/10.1007/s40314-024-02785-5 Google Scholar

[19] G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964), 4413–4416. Google Scholar

[20] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2) (2003), 417–428. https://doi.org/10.1023/A:1025407607560 Google Scholar

[21] D. V. Thong and D. V. Hieu, Mann-type algorithms for variational inequality problems and fixed point problems, Optimization 69 (10) (2020), 2305–2326. https://doi.org/10.1080/02331934.2019.1692207 Google Scholar

[22] D. V. Thong, J. Yang, Y. J. Cho, and T. M. Rassias, Explicit extragradient-like method with adaptive stepsizes for pseudomonotone variational inequalities, Optim. Lett. 15 (6) (2021), 2181–2199. https://doi.org/10.1007/s11590-020-01678-w Google Scholar

[23] P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM J. Control Optim. 38 (2) (2000), 431–446. https://doi.org/10.1137/S0363012998338806 Google Scholar