A Common Solution of Equilibrium, Constrained Convex Minimization and Fixed Point Problems
Keywords:Equilibrium problem; Constrained convex minimization problem; Averaged mapping; Iterative method; Fixed point.
In this paper, we propose a new iterative scheme with the help of the gradient- projection algorithm (GPA) for finding a common solution of an equilibrium problem, a constrained convex minimization problem, and a fixed point problem. Then, we prove some strong convergence theorems which improve and extend some recent results. Moreover, we give a numerical result to show the validity of our main theorem.
K. Aoyama, Y. Kimura, W. Takahashi and M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350-2360.
E. Blum and W. Oettli, From optimization and variatinal inequalities to equilibrium problems, Math. Student. 63(1994), 123-145.
D. P. Bretarkas and E. M. Gafin, Projection methods for variational inequalities with applications to the traffic assignment problem, Math. Program. Stud. 17(1982), 139-159.
C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Probl. 20(2004), 103-120.
K. Cheawchan, W. Phuengrattana, and A. Kangtunyakarn, A new approximation method for finding common elements of equilibrium problems, variational inequality problems and fixed point problems of nonspreading mappings, RACSAM. 111(2017), 1105-1115.
P. I. Combettes and S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Non- linear Convex Anal. 6 (2005), 117-136.
K. Geobel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math. 28, Cambridge univ. Press, 1990.
D. Han and H. K. Lo, Solving non additive traffic assignment problems: a descent method for cocoercive variational, inequalities. Eur. J. Oper. Res. 159(2004), 529-544.
Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Am. Math. Soc. 73(1967), 591-597.
S. Plubtieg and R. Punpaeng, A generaliterative method for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 336(2007), 455-469.
J. W. Peng and J. C. Yao, A viscosity approximation scheme for system of equilibrium problems, nonexpansive mappings and monotone mappings, Nonlinear Anal. (2009), doi:10.1016/j.na.2009.05.028.
A. Razani and M. Yazdi, Viscosity approximation method for equilibrium and fixed point problems, Fixed Point Theory. 14(2013), No.2, 455-472.
T. M. M. Sow, An iterative algorithm for solving equilibrium problems, variational inequal- ities and fixed point problems of multivalued quasi-nonexpansive mappings, Appl. Set- Valued Anal. Optim. 1 (2019), 171-185.
N. Shahzad and H. Zegeye, Convergence theorems of common solutions for fixed point, variational inequality and equilibrium problems, J. Nonlinear Var. Anal. 3(2019), 189-203.
W. Takahashi, Introduction to nonlinear and convex analysis, Yokohoma Publishers, Yokohoma (2009).
S. Takahashi and W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331(1)(2007), 506–515.
M. Tian and L. Liu, Iterative algorithms base on the viscosity approximation method for equilibrium and constrained convex minimization problem, Fixed Point Theory Appl. 2012:201 (2012), 1-17.
S. Wang, C. Hu, and G. Chia, Strong convergence of a new composite iterative method for equilibrium problems and fixed point problems, Appl. Math. Comput. 215 (2010), 3891- 3898.
S. Wang, Y. Zhang, W. Wang and H. Guo,Extra-gradient algorithms for split pseudomonotone equilibrium problems and fixed point problems in Hilbert spaces, J. Nonlinear Funct. Anal. 2019 , Article ID 26.
H. Xu, Averaged mappings and the gradient-projection algorithm, J. Optim. Theory Appl. 150 (2011), 360-378.
H. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116(2003), 659-678.
H. Xu, Iterative algorithms for nonlinear operators, J. Lond. Math. Soc. 66(2002), 240-256.
How to Cite
Copyright (c) 2021 Tamkang Journal of Mathematics
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.