Abstract
This paper provides an iterative construction for a common solution associated with the pseudomonotone equilibrium problems, fixed point problem of a finite family \(\eta \)-demimetric operators and the generalized split null point problem in Hilbert spaces. The sequence of approximants is a variant of the parallel shrinking extragradient algorithm with the inertial effect converging strongly to the optimal common solution under suitable set of control conditions. The viability of the approximants is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature.
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References
Alvarez, F., Attouch, H.: An inertial proximal method for monotone operators via discretization of a nonlinear oscillator with damping. Set Valued Anal. 9, 3–11 (2001)
Anh, P.N.: A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems. Bull. Malays. Math. Sci. Sco. 36(1), 107–116 (2013)
Arfat, Y., Kumam, P., Ngiamsunthorn, P.S., Khan, M.A.A.: An inertial based forward-backward algorithm for monotone inclusion problems and split mixed equilibrium problems in Hilbert spaces. Adv. Differ. Equ. 2020, 453 (2020)
Arfat, Y., Kumam, P., Ngiamsunthorn, P.S., Khan, M.A.A., Sarwar, H., Din, H.F.: Approximation results for split equilibrium problems and fixed point problems of nonexpansive semigroup in Hilbert spaces. Adv. Differ. Equ. 2020, 512 (2020)
Arfat, Y., Kumam, P., Khan, M.A.A., Ngiamsunthorn, P.S., Kaewkhao, A.: An inertially constructed forward-backward splitting algorithm in Hilbert spaces. Adv. Differ. Equ. 2021, 124 (2021)
Arfat, Y., Kumam, P., Ngiamsunthorn, P.S., Khan, M.A.A.: An accelerated projection based parallel hybrid algorithm for fixed point and split null point problems in Hilbert spaces. Math. Meth. Appl. Sci. 1–19 (2021). https://doi.org/10.1002/mma.7405
Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operators Theory in Hilbert Spaces. CMS Books in Mathematics, Springer, New York (2011)
Bauschke, H.H., Matoušková, E., Reich, S.: Projection and proximal point methods: Convergence results and counterexamples. Nonlinear Anal. 56, 715–738 (2004)
Bianchi, M., Schaible, S.: Generalized monotone bifunctions and equilibrium problems. J. Optim. Theory. Appl. 90, 31–43 (1996)
Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20, 197–228 (1967)
Byrne, C.: A unified treatment of some iterative algorithms in signal processing and image reconstruction. Inverse Probl. 20(1), 103–120 (2004)
Censor, Y., Elfving, T.: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 8, 221–239 (1994)
Censor, Y., Elfving, T., Kopf, N., Bortfeld, T.: The multiple-sets split feasibility problem and its applications. Inverse Probl. 21, 2071–2084 (2005)
Censor, Y., Gibali, A., Reich, S.: Algorithms for the split variational inequality problem. Numer. Algorithms 59, 301–323 (2012)
Combettes, P.L.: The convex feasibility problem in image recovery. Adv. Imaging Electron Phys. 95, 155–453 (1996)
Cui, H., Su, M.: On sufficient conditions ensuring the norm convergence of an iterative sequence to zeros of accretive operators. Appl. Math. Comput. 258, 67–71 (2015)
Duchi, J., Shalev-Shwartz, S., Singer, Y., Chandra, T.: Efficient projections onto the l1-ball for learning in high dimensions, In: Proceedings of the 25th International Conference on Machine Learning, Helsinki, Finland (2008)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publishers, Dordrecht (2000)
Gibali, A.: A new split inverse problem and an application to least intensity feasible solutions. Pure. Appl. Func. Anal. 2, 243–258 (2017)
Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)
Hieu, D.V., Muu, L.D., Anh, P.K.: Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings. Numer. Algorithms 73, 197–217 (2016)
Herman, G.T.: Image reconstruction from projections: the fundamentals of computerized tomography. Computer Sci. Appl. Math. (1980) (Academic Press, New York)
Khan, M.A.A.: Convergence characteristics of a shrinking projection algorithm in the sense of Mosco for split equilibrium problem and fixed point problem in Hilbert spaces. Linear Nonlinear Anal. 3, 423–435 (2017)
Khan, M.A.A., Arfat, Y., Butt, A.R.: A shrinking projection approach to solve split equilibrium problems and fixed point problems in Hilbert spaces. UPB. Sci. Bull. Ser. A 80(1), 33–46 (2018)
Korpelevich, G.M.: The extragradient method for finding saddle points and other problems. Ekon. Mat. Metody 12, 747–756 (1976)
Moudafi, A., Oliny, M.: Convergence of a splitting inertial proximal method for monotone operators. J. Comput. Appl. Math. 155, 447–454 (2003)
Polyak, B..T.: Some methods of speeding up the convergence of iteration methods. U.S.S.R. Comput. Math. Math. Phys. 4(5), 1–17 (1964)
Quoc, T.D., Muu, L.D., Hien, N.V.: Extragradient algorithms extended to equilibrium problems. Optimization 57, 749–776 (2008)
Reich, S., Tuyen, T.M.: Iterative methods for solving the generalized split common null point problem in Hilbert spaces. Optimization 69, 1013–1038 (2019)
Reich, S., Tuyen, T.M.: Parallel iterative methods for solving the generalized split common null point problem in Hilbert spaces. RACSAM 114, 180 (2020)
Tada, A., Takahashi, W.: Strong Convergence Theorem for an Equilibrium Problem and a Nonexpansive Mapping. Yokohama Publishers, Yokohama, Nonlinear Anal. Conv. Anal (2006)
Takahashi, W.: Nonlinear Functional Analysis: Fixed Point Theory and its Applications. Yokohama Publishers, Yokohama (2000)
Takahashi, S., Takahashi, W.: The split common null point problem and the shrinking projection method in Banach spaces. Optimization 65, 281–287 (2016)
Takahashi, W.: The split common fixed point problem and the shrinking projection method in Banach spaces. J. Conv. Anal. 24, 1015–1028 (2017)
Takahashi, W.: Strong convergence theorem for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Japan J. Indust. Appl. Math. 34, 41–57 (2017)
Takahashi, W.: Weak and strong convergence theorems for new demimetric mappings and the split common fixed point problem in Banach spaces. Numer. Funct. Anal. Optim. 39(10), 1011–1033 (2018)
Takahashi, W., Wen, C.F., Yao, J.C.: The shrinking projection method for a finite family of demimetric mappings with variational inequality problems in a Hilbert space. Fixed Point Theory 19, 407–419 (2018)
Tiel, J.: Convex Analysis. An Introductory Text. Wiley, Chichester (1984)
Vogel, C.R.: Computational Methods for Inverse Problems. SIAM, Philadelphia (2002)
Acknowledgements
The authors wish to thank the anonymous referees for their comments and suggestions. The authors (Y. Arfat, P. kumam and P. S. Ngiamsunthorn) acknowledge the financial support provided by the Center of Excellence in Theoretical and Computational Science (TaCS-CoE), KMUTT. Moreover, this research project is supported by Thailand Science Research and Innovation (TSRI) Basic Research Fund: Fiscal year 2021 under project number 64A306000005. The author Yasir Arfat was supported by the Petchra Pra Jom Klao Ph.D Research Scholarship from King Mongkut’s University of Technology Thonburi, Thailand (Grant No.16/2562). Also, this project was funded by National Council of Thailand (NRCT) under Research Grants for Talented Mid-Career Researchers (Contract no. N41A640089).
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Arfat, Y., Kumam, P., Khan, M.A.A. et al. Parallel shrinking inertial extragradient approximants for pseudomonotone equilibrium, fixed point and generalized split null point problem. Ricerche mat 73, 937–963 (2024). https://doi.org/10.1007/s11587-021-00647-4
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DOI: https://doi.org/10.1007/s11587-021-00647-4
Keywords
- Shrinking approximants
- Pseudomonotone equilibrium problem
- Fixed point problem
- Demimetric operator
- Generalized null point problem