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

Year 2019, Volume: 2 Issue: 3, 102 - 112, 01.10.2019

Caifen Zhang Jinzuo Chen

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

Supporting Institution

Lingnan Normal University

Project Number

1171518004,2018A0303070012,2017KQNCX125

Thanks

This work was supported by the Key Subject Program of Lingnan Normal University (1171518004), the Natural Science Foundation of Guangdong Province (2018A0303070012) and the Young Innovative Talents Project in Guangdong Universities (2017KQNCX125).

References

  • [1] Stampacchia, G: Forms bilineaires coercivities sur les ensembles convexes. CR Acad Sci Paris. 258, 4413-4416(1964)
  • [2] Pitea, A: Results on some classes of vector variational inequalities. J. Nonlinear Convex Anal. 19, 417-432(2018)
  • [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] 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] 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] 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] Yao, J., Zheng, X: Existence of solutions and error bound for vector variational inequalities in Banach spaces. Optimization.67, 1333-1344(2018)
  • [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] Ceng, L., Wen, C: Relaxed extragradient methods for systems of variational inequalities. J. Inequal. Appl. 2015, Article ID140(2015)
  • [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] 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] Korpelevich, G: An extra gradient method for finding saddle points and for other problems. Ekonomika Mat. Metody. 12,747-756(1976)
  • [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] 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] Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear.Anal. 70, 4039-4046(2009)
  • [16] Yao, Y., Chen, R., Yao, J: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal.68, 1687-1693(2008)
  • [17] Yao, Y., Liou, Y., Yao, J: Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithmconstructi

Year 2019, Volume: 2 Issue: 3, 102 - 112, 01.10.2019

Caifen Zhang Jinzuo Chen

Abstract

Project Number

1171518004,2018A0303070012,2017KQNCX125

References

  • [1] Stampacchia, G: Forms bilineaires coercivities sur les ensembles convexes. CR Acad Sci Paris. 258, 4413-4416(1964)
  • [2] Pitea, A: Results on some classes of vector variational inequalities. J. Nonlinear Convex Anal. 19, 417-432(2018)
  • [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] 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] 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] 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] Yao, J., Zheng, X: Existence of solutions and error bound for vector variational inequalities in Banach spaces. Optimization.67, 1333-1344(2018)
  • [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] Ceng, L., Wen, C: Relaxed extragradient methods for systems of variational inequalities. J. Inequal. Appl. 2015, Article ID140(2015)
  • [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] 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] Korpelevich, G: An extra gradient method for finding saddle points and for other problems. Ekonomika Mat. Metody. 12,747-756(1976)
  • [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] 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] Zhou, H: Strong convergence of an explicit iterative algorithm for continuous pseudo-contractions in Banach spaces. Nonlinear.Anal. 70, 4039-4046(2009)
  • [16] Yao, Y., Chen, R., Yao, J: Strong convergence and certain control conditions for modified Mann iteration. Nonlinear Anal.68, 1687-1693(2008)
  • [17] Yao, Y., Liou, Y., Yao, J: Split common fixed point problem for two quasi-pseudo-contractive operators and its algorithmconstructi
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Caifen Zhang This is me

Jinzuo Chen This is me

Project Number 1171518004,2018A0303070012,2017KQNCX125
Publication Date October 1, 2019
Published in Issue Year 2019 Volume: 2 Issue: 3

Cite

APA Zhang, C., & Chen, J. (2019). Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. Results in Nonlinear Analysis, 2(3), 102-112.
AMA Zhang C, Chen J. Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. RNA. October 2019;2(3):102-112.
Chicago Zhang, Caifen, and Jinzuo Chen. “Convergence Analysis of Variational Inequality and Fixed Point Problems for Pseudo-Contractive Mapping With Lipschitz Assumption”. Results in Nonlinear Analysis 2, no. 3 (October 2019): 102-12.
EndNote Zhang C, Chen J (October 1, 2019) Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. Results in Nonlinear Analysis 2 3 102–112.
IEEE C. Zhang and J. Chen, “Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption”, RNA, vol. 2, no. 3, pp. 102–112, 2019.
ISNAD Zhang, Caifen - Chen, Jinzuo. “Convergence Analysis of Variational Inequality and Fixed Point Problems for Pseudo-Contractive Mapping With Lipschitz Assumption”. Results in Nonlinear Analysis 2/3 (October 2019), 102-112.
JAMA Zhang C, Chen J. Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. RNA. 2019;2:102–112.
MLA Zhang, Caifen and Jinzuo Chen. “Convergence Analysis of Variational Inequality and Fixed Point Problems for Pseudo-Contractive Mapping With Lipschitz Assumption”. Results in Nonlinear Analysis, vol. 2, no. 3, 2019, pp. 102-1.
Vancouver Zhang C, Chen J. Convergence analysis of variational inequality and fixed point problems for pseudo-contractive mapping with Lipschitz assumption. RNA. 2019;2(3):102-1.