Convergence of Common Solution of Variational Inequality and Fixed Point of a Pseudocontractive Mapping

  • Poonam MishraEmail author
Conference paper


Variational inequalities arise in models for a large number of mathematical, physical, economics, finance, optimization, game theory, engineering, and other problems. The fixed point formulation of any variational inequality problem is not only useful for the existence of solution of the variational inequality problem, but it also provides the facility to develop algorithms for the approximation of solution of variational inequality problem. A lot of research has been carried out to approximate common solution of fixed point of nonexpansive mapping and solution of a variational inequality. In this paper, we propose an iterative algorithm which gives a common element of fixed point of asymptotically pseudocontractive mapping and a certain variational inequality. Our result extends the previously known results from Hilbert Space to Banach Space and from asymptotically nonexpansive mapping to asymptotically pseudocontractive mapping.


Strong convergence Asymptotically pseudocontractive mapping Nonexpansive mapping Variational inequality 

AMS Subject Classification

49H09 47H10 47J20 49J40 


  1. 1.
    Xu, H.K., Viscosity approximation methods for nonexpansive mappings, J. Math. Anal. Appl. 2004;298: 279–29.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Nakajo, K., Takahashi, W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 2003;279: 372–379.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Xu, H.K., Alghamdi, M.Ali, Shahzad, N., The viscosity technique for the implicit midpoint rule of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications 2015:41.Google Scholar
  4. 4.
    Lou, J., Zhang, L., He, Z., Viscosity approximation methods for asymptotically nonexpansive mappings, Appl. Math. Comput. 2008, 203:171–177.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Moudafi, A. Viscosity approximation methods for fixed points problems, J. Math. Anal. 241 2000; 241: 46–55.MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bader, G., deuflhard, P: A semi-implicit mid-point rule for stiff systems of ordinary differential equations, Numer. Math. 1983;41: 373–398.MathSciNetCrossRefGoogle Scholar
  7. 7.
    Deuflhard, P: Recent progress in extrapolation methods for ordinary differential equations., SIAM Rev. 1985; 27(4):505–535.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Alghamadi, M.A., Alghamdi, M. Ali., Shahzad, N., Xu, H.K. The implicit midpoint rule for nonexpansive mappings, Fixed Point Theory and Applications 2014:96.Google Scholar
  9. 9.
    Luo, Ping., Cai, Gang, Shehu, Yekini. The viscosity iterative algorithms for the implicit midpoint rule of nonexpansive mappings in uniformly smooth Banach spaces, Journal of Inequalities and Applications (2017) 2017:154.Google Scholar
  10. 10.
    Yan, Q., Hu, Shaotao: Strong convergence theorems for the generalized viscosity implicit rules of asymptotically nonexpansive mappings in Hilbert spaces. J. Computational Analysis and Applications, 2018; 24(3):486–496.Google Scholar
  11. 11.
    Wang, Y.H., Xia, Y.H., Strong Convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces, Fixed Point Theory Appl., 2012, 8 pages, 1.2.Google Scholar
  12. 12.
    Yao, Y.H, Shahzad, N., Liou, Y.,C., Modified semi-implicit midpoint rule for nonexpansive mappings, Fixed Point Theory Appl., 2015, 15 pages,1,1.3,2.5 theory and Applications 2014:96.Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Amity School of Engineering and TechnologyAmity UniversityRaipurIndia

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