Nonlinear Functional Analysis and Applications

Kyungnam University, Department of Mathematics Eduaction (경남대학교 수학교육과)
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ACCELERATED STRONGLY CONVERGENT EXTRAGRADIENT ALGORITHMS TO SOLVE VARIATIONAL INEQUALITIES AND FIXED POINT PROBLEMS IN REAL HILBERT SPACES

  • Nopparat Wairojjana (Applied Mathematics Program, Faculty of Science and Technology, Valaya Alongkorn Rajabhat University under the Royal Patronage Pathum Thani Province) ;
  • Nattawut Pholasa (School of Science, University of Phayao) ;
  • Chainarong Khunpanuk (Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University) ;
  • Nuttapol Pakkaranang (Mathematics and Computing Science Program, Faculty of Science and Technology, Phetchabun Rajabhat University)
  • Received : 2022.12.22
  • Accepted : 2024.04.14
  • Published : 2024.06.15
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Abstract

Two inertial extragradient-type algorithms are introduced for solving convex pseudomonotone variational inequalities with fixed point problems, where the associated mapping for the fixed point is a 𝜌-demicontractive mapping. The algorithm employs variable step sizes that are updated at each iteration, based on certain previous iterates. One notable advantage of these algorithms is their ability to operate without prior knowledge of Lipschitz-type constants and without necessitating any line search procedures. The iterative sequence constructed demonstrates strong convergence to the common solution of the variational inequality and fixed point problem under standard assumptions. In-depth numerical applications are conducted to illustrate theoretical findings and to compare the proposed algorithms with existing approaches.

Keywords

Acknowledgement

The first author would like to thank Faculty of Science and Technology and Research and Development Institute, Valaya Alongkorn Rajabhat University under the Royal Patronage Pathun Thani Province. The second author was supported by University of Phayao and Thailand Science Research and Innovation Fund (Fundamental Fund 2024). The fourth author would like to thank Professor Dr. Poom Kumam from King Mongkuts University of Technology Thonburi, Thailand for his advice and comments to improve the results of this paper. This research (Grant No. RGNS 65-168) was supported by Office of the Permanent Secretary, Ministry of Higher Education, Science, Research and Innovation (OPS MHESI), Thailand Science Research and Innovation (TSRI) and Phetchabun Rajabhat University.

References

  1. H.A. Abass, O.K. Narain and O.M. Onifade, Inertial extrapolation method for solving systems of monotone variational inclusion and fixed point problems using Bregman distance approach , Nonlinear Funct. Anal. Appl., 28(2) (2023), 497-520. 
  2. K. Abuasbeh, A. Kanwal, R. Shafqat, B. Taufeeq, M.A. Almulla and M. Awadalla, A method for solving time-fractional initial boundary value problems of variable order, Symmetry, 15(2) (2023), Article ID 519. 
  3. K. Abuasbeh and R. Shafqat, Fractional brownian motion for a system of fuzzy fractional stochastic differential equation, J. Math., 2022 (2022), Artcle ID 3559035. 
  4. N.T. An, N.M. Nam and X. Qin, Solving k-center problems involving sets based on optimization techniques, J. Global Optim., 76(1) (2019), 189-209. 
  5. A.S. Antipin, On a method for convex programs using a symmetrical modification of the Lagrange function, Economika i Matem. Metody, 12 (1976), 1164-1173. 
  6. L.C. Ceng, A. Petru,sel, X. Qin and J.C. Yao, A modified inertial subgradient extragradient method for solving pseudomonotone variational inequalities and common fixed point problems, Fixed Point Theory, 21(1) (2020), 93-108. 
  7. L.C. Ceng, J.C. Yao and Y. Shehu, On mann implicit composite subgradient extragradient methods for general systems of variational inequalities with hierarchical variational inequality constraints, J. Ineq. Appl., 2022 (2022), Article ID 78. 
  8. C.M. Elliott, Variational and quasivariational inequalities: applications to free-boundary problems (Claudio Baiocchi and antonio Capelo), SIAM Rev., 29(2) (1987), 314-315. 
  9. H.A. Hammad, H. Rehman and M.D. la Sen, Advanced algorithms and common solutions to variational inequalities, Symmetry, 12 (2020), Article ID 1198. 
  10. H. Iiduka and I. Yamada, A subgradient-type method for the equilibrium problem over the fixed point set and its applications, Optimization, 58(2) (2009), 251-261. 
  11. G. Kassay, J. Kolumban and Z. Pales, On nash stationary points, Publ. Math. Debrecen, 54(3-4) (1999), 267-279. 
  12. G. Kassay, J. Kolumban and Z. Pales, Factorization of minty and stampacchia variational inequality systems, Euro. J. Oper. Res., 143 (2002), 377-389. 
  13. A. Khan, R. Shafqat and A.U.K. Niazi, Existence results of fuzzy delay impulsive fractional differential equation by fixed point theory approach, J. Funct. Spaces, 2022 (2022), Article ID 4123949. 
  14. J.K. Kim, A.H. Dar and Salahuddin, Existence theorems for the generalized relaxed pseudomonotone variational inequalities, Nonlinear Funct. Anal. Appl., 25(1) (2020), 25-34. 
  15. D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Classic in Applied Mathematics, Society for Industrial and Applied Mathematics, (SIAM), Philadelphia, PA, 2000. 
  16. I. Konnov, Equilibrium models and variational inequalities, 210, Elsevier, 2007. 
  17. G. Korpelevich, The extragradient method for finding saddle points and other problems, Ekonom. i Mat. Metody, 12(4) (1976), 747-756. 
  18. R. Kraikaew and S. Saejung, Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces, J. Optim. Theory Appl., 163(2) (2014), 399-412. 
  19. G.M. Lee and J.H. Lee, On Augmented Lagrangian Methods of Multipliers and Alternating Direction Methods of Multipliers for Matrix Optimization Problems , Nonlinear Funct. Anal. Appl., 27 (4) (2022), 869-879. 
  20. P.E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal., 16 (2008), 899-912. 
  21. P.E. Mainge and A. Moudafi, Coupling viscosity methods with the extragradient algorithm for solving equilibrium problems, J. Nonlinear Convex Anal., 9(2) (2008), 283-294. 
  22. A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46-55. 
  23. A. Moumen, R. Shafqat, Z. Hammouch, A.U.K. Niazi and M.B. Jeelani, Stability results for fractional integral pantograph differential equations involving two caputo operators, AIMS Math., 8(3) (2023), 6009-6025. 
  24. K. Muangchoo, A Viscosity type Projection Method for Solving Pseudomonotone Variational Inequalities, Nonlinear Funct. Anal. Appl., 26(2) (2021), 347-371. 
  25. A. Nagurney, Network economics: a variational inequality approach, Kluwer Academic Publishers Group, Dordrecht, 1993, https://doi.org/10.1007/978-94-011-2178-1. 
  26. M.A. Noor, Some iterative methods for nonconvex variational inequalities, Math. Comput. Modelling, 54 (11-12) (2011), 2955-2961. 
  27. X. Qin and N.T. An, Smoothing algorithms for computing the projection onto a Minkowski sum of convex sets, Comput. Optim. Appl., 74(3) (2019), 821-850. 
  28. S. Saejung and P. Yotkaew, Approximation of zeros of inverse strongly monotone operators in Banach spaces , Nonlinear Anal., 75(2) (2012), 742-750. 
  29. Y. Shehu, Q.L. Dong and D. Jiang, Single projection method for pseudo-monotone variational inequality in Hilbert spaces, Optimization, 68(1) (2018), 385-409. 
  30. G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris, 258 (1964), 4413-4416. 
  31. W. Takahashi, Introduction to nonlinear and convex analysis, Yokohama, 2009. 
  32. B. Tan, Z. Zhou and S. Li, Viscosity-type inertial extragradient algorithms for solving variational inequality problems and fixed point problems, J. Appl. Math. Comput., 68(2) (2022), 1387-1411. 
  33. I. Yamada and N. Ogura, Hybrid steepest descent method for variational inequality problem over the fixed point set of certain quasi-nonexpansive mappings, Numer. Funct. Anal. Optim., 25(7-8) (2004), 619-655. 
  34. P. Yordsorn, P. Kumam and H.U. Rehman, Modified two-step extragradient method for solving the pseudomonotone equilibrium programming in a real Hilbert space, Carpathian J. Math., 36(2) (2020), 313-330. 
  35. P. Yordsorn, P. Kumam, H.U. Rehman and A.H. Ibrahim, A weak convergence self-adaptive method for solving pseudomonotone equilibrium problems in a real Hilbert space, Mathematics, 8(7) (2020), Article ID 1165.