International Journal of Nonlinear Analysis and Applications

A convergence theorem for a common solution of $f$-fixed point, variational inequality and generalized mixed equilibrium problems in Banach spaces

Document Type : Research Paper

Authors

1 Department of Mathematics, Faculty of Science, University of Botswana, Pvt Bag 00704, Gaborone, Botswana

2 Department of Mathematics and Statistical Sciences, Faculty of Sciences, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana

Abstract

The purpose of this paper is to construct an algorithm for approximating a common element of the set of solutions of a finite family of generalized mixed equilibrium problems, the set of $f$-fixed points of a finite family of $f$-pseudocontractive mappings, and the set of solutions of a finite family of variational inequality problems for Lipschitz monotone mappings in real reflexive Banach spaces.

Keywords

[1] H.H. Bauschke, J.M. Borwein and P.L. Combettes, Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces, Commun. Contemp. Math. 3 (2001), 615–647.
[2] A.U. Bello and M.O. Nnakwe, An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application, Adv. Differ. Eq. 2021 (2021), no. 1, 1–17.
[3] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145.
[4] F.J. Bonnans and A. Shapiro, Perturbation analysis of optimization problem, Springer, New York, 2000.
[5] L.M. Bregman, The relaxation method for finding common points of convex sets and its application to the solution of problems in convex programming, USSR Comput. Math. Math. Phys. 7 (1967), 200–217.
[6] D. Butnariu and E. Resmerita, Bregman distances, totally convex functions and a method for solving operator equations in Banach spaces, Abstr. Appl. Anal. 2006 (2006), 139.
[7] F.E. Browder, Existence and approximation of solutions of nonlinear variational inequalities, Proc. Natl. Acad. Sci. USA 56 (1966), no. 4, 1080–1086.
[8] L.C. Ceng and J.C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186–201.
[9] V. Darvish, Strong convergence theorem for generalized mixed equilibrium problems and Bregman nonexpansive mapping in Banach spaces, Mathematica Moravica 20 (2016), no. 1, 69–87.
[10] M. Khonchaliew, A. Farajzadeh and N. Petrot, Shrinking extra gradient method for pseudo monotone equilibrium problems and quasi-nonexpansive mappings, Symmetry 11 (2019), no. 4, 480.
[11] P. Lohawech, A. Kaewcharoen and A. Farajzadeh, Algorithms for the common solution of the split variational inequality problems and fixed point problems with applications, J. Inequal. Appl. 2018 (2018), 358.
[12] P.E. Maing´e, Strong convergence of projected subgradient method for nonsmooth and nonstrictily convex minimization, Set-Valued Anal. 16 (2008), 899–912.
[13] A. Mouda and M. Thera, Proximal and dynamical approaches to equilibrium problems, Lecture notes in Economics and Mathematical Systems, 477, Springer, 1999, 187–201.
[14] M.O. Nnakwe and C.C. Okeke, A common solution of generalized equilibrium problems and fixed points of pseudo contractive-type maps, J. Appl. Math. Comput. 66 (2021), no. 1, 701–716.
[15] R.P. Phelps, Convex functions, monotone operators, and differentiability, Lecture Notes in Mathematics, vol. 1364, 2nd edn. Springer, Berlin 1993.
[16] S. Reich, Product formulas, nonlinear semigroups, and accretive operators, J. Funct. Anal. 36 (1980), 147–168.
[17] S. Reich and S. Sabach, Strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces, J. Nonlinear Convex Anal. 10 (2009), 471–485.
[18] S. Reich and S. Sabach, Two strong convergence theorems for a proximal method in reflexive Banach spaces, Numer. Funct. Anal. Optim. 31 (2010), 22–44.
[19] R.T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim. 14 (1976), no. 5, 877–898.
[20] P. Senakka and P. Cholamjiak, Approximation method for solving fixed point problem of Bregman strongly nonexpansive mappings in reflexive Banach spaces, Ric. Mat. 65 (2016), 209–220.
[21] N. Shahzad and H. Zegeye, Convergence theorems of common solutions for fixed point, variational inequality and equilibrium problems, J. Nonlinear Var. Anal. 3 (2019), no. 2, 189–203.
[22] H.K. Xu, Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 298 (2004), no. 1, 279–291.
[23] G.B. Wega and H. Zegeye, Convergence results of Forward-Backward method for a zero of the sum of maximally monotone mappings in Banach spaces, Comput. Appl. Math. 39 (2020), 223.
[24] H. Zegeye, An iterative approximation for a common fixed point of two pseudo-contractive mappings, Int. Scholar. Res. Notices 2011 (2011).
[25] H. Zegeye and G.B. Wega, Approximation of a common f-fixed point of f-pseudo contractive mappings in Banach spaces, Rend. Circ. Mat. Palermo (2) 70 (2021), no. 3, 1139–1162
International Journal of Nonlinear Analysis and Applications
Volume 13, Issue 2
July 2022
Pages 1069-1087
  • Receive Date: 27 November 2021
  • Revise Date: 23 March 2022
  • Accept Date: 01 April 2022