Viscosity approximation method for a variational problem

Volume 16, Issue 4, pp 208--221 http://dx.doi.org/10.22436/jnsa.016.04.02

Publication Date: October 26, 2023 Submission Date: May 05, 2023 Revision Date: September 05, 2023 Accteptance Date: September 23, 2023

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 212 Downloads
• 591 Views

Authors

R. May - Mathematics Department, College of Science, King Faisal University, P.O. 380, Ahsaa 31982, Kingdom of Saudi Arabia.

Abstract

By combining the works of Moudafi [A. Moudafi, J. Math. Anal. Appl., \(\textbf{241}\) (2000), 46--55] and Iiduka and Takahashi [H. Iiduka, W. Takahashi, Nonlinear Anal., \(\textbf{61}\) (2005), 341--350], we introduce an iterative process that converges strongly to a particular solution of a variational inequality problem. We also study the stability of the algorithm under relatively small perturbation and we apply the obtained results to the study of a constrained optimization problem and a problem of common fixed points of two nonexpansive mappings. Some numerical experiments are provided to study the affect of some parameters on the speed of the convergence of the considered algorithm.

Share and Cite

Keywords

• Hilbert spaces
• variational inequality problem
• nonexpansive mapping
• inverse strongly monotone mappings

MSC

•  47H09
•  47H05
•  47H10
•  47J20

References

• [1] H. H. Bauschke, P. L. Combettes, Convex analysis and monotone operator theory in Hilbert spaces, In: CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, Springer, New York (2011)
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [2] D. P. Bertsekas, Nonlinear programming, J. Oper. Res. Soc., 48 (1997), 334–334
  • View Article


• [3] P. Cholamjiak, S. Kesornprom, N. Pholasa, Weak and strong convergence theorems for the inclusion problem and the fixed-point problem of nonexpansive mapping, Mathematics, 7 (2019), 19 pages
  • View Article
  • Google Scholar


• [4] O. G¨ uler, Foundations of optimization, Springer, New York (2010)
  • View Article
  • MathSciNet
  • Google Scholar


• [5] H. Iiduka, W. Takahashi, Strong convergence theorems for nonexpansive mapping and inverse-strongly monotone mappings, Nonlinear Anal., 61 (2005), 341–350
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] A. Moudafi, Viscosity approximation methods for fixed-points problems, J. Math. Anal. Appl., 241 (2000), 46–55
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] J. Peypoquet, Convex optimization in normed spaces: theory, methods and examples, Springer Briefs Optim., Springer, Cham (2015)
  • View Article
  • Google Scholar
  • MATH


• [8] S. Plubteing, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math., 224 (2009), 614–621
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [9] W. Takahashi, M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl., 118 (2003), 417–428
  • View Article
  • MathSciNet
  • Google Scholar


• [10] G. C. Ugwunnadi, Viscosity approximation method for solving variational inequality problem in real Banach spaces, Armen. J. Math., 13 (2021), 1–20
  • View Article
  • MathSciNet
  • Google Scholar


• [11] Y. Wang, C. Pan, Viscosity approximation methods for a general variational inequality system and fixed point problems in Banach spaces, Symmetry, 12 (2020), 1–15
  • View Article
  • Google Scholar


• [12] H.-K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. (2), 66 (2002), 240–256
  • View Article
  • MathSciNet
  • Google Scholar


• [13] H.-K. Xu, Averaged mapping and the gradient-projection algorithm, J. Optim. Theory Appl., 150 (2011), 360–378
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [14] Y. Yao, Y.-C. Liou, C.-P. Chen, Algorithms construction for nonexpansive mappings and inverse-strongly monotone mappings, Taiwanese J. Math., 15 (2011), 1979–1998
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] M. Yazdi, S. H. Sababe, A viscosity approximation methods for solving general system of variational inequalities, generalized mixed equilibrium problems and fixed point problems, Symmetry, 14 (2022), 1–23
  • View Article
  • Google Scholar