Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption

Abstract

In this paper, we consider and study variational inequality and fixed point problems for pseudo-contractive mapping. It is proven that the sequences generated by the proposed iterative algorithm converge strongly to the common solution of the variational inequality and fixed point problems. A numerical example illustrates the theoretical result.

Keywords

• variational inequality problem,fixed point problem,extra-gradient method,pseudo-contractive mapping,strong convergence

Kaynakça

1. [1] Stampacchia, G: Forms bilineaires coercivities sur les ensembles convexes. CR Acad Sci Paris. 258, 4413-4416(1964)
2. [2] Pitea, A: Results on some classes of vector variational inequalities. J. Nonlinear Convex Anal. 19, 417-432(2018)
3. [3] Nadezhkina, N., Takahashi, W: Weak convergence theorem by an extragradient method for nonexpansive mappings andmonotone mappings. J. Optimiz. Theory. App. 128, 191-201(2006)
4. [4] Yao, Y., Postolache, M., Yao, J: Iterative algorithms for generalized variational inequalities. Politehn. Univ. Bucharest Sci.Bull. Ser. A Appl. Math. Phys. 81, 3-16(2019)
5. [5] Ceng, L., Petrusel, A., Yao, J., Yao, Y: Systems of variational inequalities with hierarchical variational inequality constraintsfor Lipschitzian pseudocontractions. Fixed Point Theory. 20, 113-133(2019)
6. [6] Ceng, L., Yao, J., Yao, Y: Existence of solutions for a class of variational-hemivariational-like inequalities in Banach spaces.Filomat. 32, 3609-3622(2018)
7. [7] Yao, J., Zheng, X: Existence of solutions and error bound for vector variational inequalities in Banach spaces. Optimization.67, 1333-1344(2018)
8. [8] Petrusel, A., Petrusel, G., Yao, J: Variational analysis concepts in the theory of multi-valued coincidence problems. J.Nonlinear Convex Anal. 19, 935-958(2018)

9. [9] Ceng, L., Wen, C: Relaxed extragradient methods for systems of variational inequalities. J. Inequal. Appl. 2015, Article ID140(2015)
10. [10] Yao, Y., Liou, Y., Yang, P: Coupling extra-gradient methods with KM’s methods for variational inequalities and fixedpoints. Taiwan. J. Math. 16, 1329-1343(2012)
11. [11] Kangtunyakarn, A: The modification of the system of variational inequalities for fixed point theory in Banach spaces. FixedPoint Theory Appl. 2014, Article ID 123(2014)
12. [12] Korpelevich, G: An extra gradient method for finding saddle points and for other problems. Ekonomika Mat. Metody. 12,747-756(1976)
13. [13] Bertsekas, D., Gafni, E: Projection methods for variational inequalities with applications to the traffic assignment problem.Math. Program. Stud. 17, 139-159(1982)
14. [14] Han, D., Lo, H: Solving non-additive traffic assignment problems: a descent method for co-coercive variational inequalities.European J. Oper. Res. 159, 529-544(2004)
15. [15] Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear.Anal. 70, 4039-4046(2009)
16. [16] Yao, Y., Chen, R., Yao, J: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal.68, 1687-1693(2008)
17. [17] Yao, Y., Liou, Y., Yao, J: Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithmconstructi