Hybrid extragradient-type algorithm for zeros and fixed point problems in Banach spaces
محورهای موضوعی : Fixed point theory
B. Ali
1
(Department of Mathematical Sciences, Bayero University, Kano, Nigeria)
A. A. Alasan
2
(Department of Mathematical Sciences, Bayero University, Kano, Nigeria; Department of Basic Studies, Kano State Polytechnic, Kano, Nigeria)
کلید واژه: equilibrium problem, maximal monotone operator, Bregman inverse strongly monotone operator, Bregman demigeneralized mapping,
چکیده مقاله :
In this paper, we introduce a new hybrid extragradient-type algorithm for approximating an element in the set of common solutions of equilibrium problems and common fixed points of family of Bregman demigeneralized mappings which is also a common zero of the sum of maximal monotone and Bregman inverse strongly monotone operators in the setting of reflexie Banach space. Strong convergence of the proposed algorithm to a solutions of the said problems is established which improves and generalizes many recently announced results in the literature.
In this paper, we introduce a new hybrid extragradient-type algorithm for approximating an element in the set of common solutions of equilibrium problems and common fixed points of family of Bregman demigeneralized mappings which is also a common zero of the sum of maximal monotone and Bregman inverse strongly monotone operators in the setting of reflexie Banach space. Strong convergence of the proposed algorithm to a solutions of the said problems is established which improves and generalizes many recently announced results in the literature.
[1] B. Ali, G. C. Ugwunnadi, M. S. Lawan, V. Darvish, Solution of generalized mixed equilibrium and common fixed point problems for Bregman demigeneralized mappings, J. Analysis. 31 (2023), 2203-2224.
[2] H. H. Bauschke, J. M. Borwein, P. L. Combettes, Bregman monotone optimization algorithms, SIAM J. Control. Optim. 42 (2003), 596-636.
[3] H. H. Bauschke, P. L. Combettes, J. M. Borwein, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615-647.
[4] E. Blum, W. Oettli, From optimazation and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123-145.
[5] J. F. Bonnans, A. Shapiro, Perturbation Analysis of Optimization Problems, New York, Springer Verlag, 2000.
[6] L. M. Bregman, The relaxation method for finding the common point of convex set and its application to solution of convex programming, USSR Comput. Matt. & Math. Phys. 7 (1967), 200-217.
[7] D. Butnariu, A. N. Iusem, Totally Convex Functions for Fixed Points Computation and Infinite Dimensional Optimization, Kluwer Academic Publishers, Dordrecht, 2000.
[8] D. Butnariu, G. Kassay, A proximal-projection method for finding zeros of set-valued operators, SIAM J. Control. Optim. 47 (2008), 2096-2136.
[9] D. Butnariu, E. Resmerita, Bregman distances, totally convex functions and a method of solving operator equations in Banach spaces, Abstr. Appl. Anal. (2006), 2006:084919.
[10] Y. Censor, A. Lent, An iterative row-action method interval convex programming, J. Optim. Theo. Appl. 34 (1981), 321-353.
[11] I. Cioranescu, Geometry of Banach Spaces: Duality mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.
[12] G. Z. Eskandani, M. Raeisi, T. M. Rassias, A hybrid extragradient method for solving pseudomonotone equilibrium problems using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 20:132.
[13] S. Kamimura, W. Takahashi, Approximating solutions of maximal monotone operators in Hilbert spaces, J. Approx. Theory. 106 (2) (2000), 226-240.
[14] R. T. Kacurovskii, Monotone operators and convex functionals, Uspekhi Mat. Nauk. 15 (1960), 213-215.
[15] F. Kohsaka, W. Takahashi, Proximal point algorithms with Bregman functions in Banach spaces, J. Nonlinear Convex Anal. 6 (2005), 505-523.
[16] P. L. Lions, B. Mercier, Splitting algorithms for the sum of two nonlinear operators, SIAM J. Numer. Anal. 16 (1979), 964-979.
[17] P. E. Mainge, Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Anal. 16 (2008), 899-912.
[18] G. J. Minty, Monotone (nonlinear) operators in Hilbert space, Duke Math. J. 29 (1962), 341-346.
[19] E. Naraghirad, J. C. Yao, Bregman weak reletively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2013), 2013:141.
[20] F. U. Ogbuisi, I. O. Jolaoso, F. O. Isiogugu, A Halpern-type iteration method for Bregman nonspreading mappings and monotone operators in reflexive Banach space, J. Math. (2019), 2019: 8059135.
[21] G. B. Passty, Ergodic convergence to a zero of the sum of monotone operators in Hilbert space, J. Math. Anal. Appl. 72 (20) (1979), 383-390.
[22] R. P. Phelps, Convex Functions, Monotone Operators, and Differentiability, Lecture notes in mathematics, Berlin, Springer, 1993.
[23] T. D. Quoc, L. D. Muu, V. H. Nguyen, Extragradient methods extended to equilibrium problems, Optimization. 57 (2008), 749-776.
[24] S. Reich, A Weak Convergence Theorem for the Alternating Method with Bregman Distances, CRC Press, Boca Raton, FL, USA, 1996.
[25] S. Reich, S. Sabach, A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471-485.
[26] S. Reich, S. Sabach, Existence and Approximation of Fixed Points of Bregman Firmly Nonexpansive Mappings in Reflexive Banach Spaces Springer, 2011.
[27] S. Reich, S. Sabach, Two strong convergence theorems for Bregman strongly nonexpansive operator in reflexive Banach space, Nonlinear Anal. 13 (2010), 122-135.
[28] S. Reich, S. Sabach, Two strong convergence theorems for proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 22-44.
[29] J. V. Tiel, Convex Analysis: An Introductory Text, Wiley, New York, 1984.
[30] T. M. Tuyen, R. Promkam, P. Sunthrayuth, Strong convergence of a generalized forward-backward splitting method in reflexive Banach spaces, Optimization. 71 (6) (2022), 1483-1508.
[31] C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing, Singapore, 2002.
[32] E. H. Zarantonello, Solving Functional Equations by Contractive Averaging, Mathematical Research Center Technical Summary Report, University of Wisconsin, Madison, 160, 1960.
