Abstract

We apply fixed point iterative schemes to variational inequality problems, via admissible perturbations of projection operators in real Hilbert spaces. Then, we prove some convergence theorems, extending and complementing the results in the existing literature. In particular, we deal with the class of α-co-coercive operators with application to general equilibrium problems.

Author information

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Elena Toscano

Dept. of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

elena.toscano@unipa.it

Calogero Vetro

Dept. of Mathematics and Computer Sciences, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy

calogero.vetro@unipa.it

Keywords

  • Hilbert space
  • Krasnoselskij-type iterative scheme
  • projected dynamical system
  • projection operator
  • variational inequality problem

Mathematics Subject Classification

41A65, 47H10

References

  1. R. P. Agarwal, D. O’Regan, D. R. Sahu: Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, New York (2009).Google Scholar
  2. R. P. Agarwal, D. O’Regan, D. R. Sahu: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, J. Nonlinear Convex Anal. 8 (2007) 61–79.Google Scholar
  3. V. Berinde: Convergence theorems for admissible perturbations of φ-pseudocontractive operators, Miskolc Math. Notes 15 (2014) 27–37.https://doi.org/10.18514/MMN.2014.1003Google Scholar
  4. V. Berinde: Iterative Approximation of Fixed Points, 2nd edition, Springer, Berlin (2007).https://doi.org/10.1109/SYNASC.2007.49Google Scholar
  5. V. Berinde: Convergence theorems for fixed point iterative methods defined as admissible perturbations of a nonlinear operator, Carpathian J. Math. 29 (2013) 9–18.https://doi.org/10.37193/CJM.2013.01.15Google Scholar
  6. V. Berinde, G. Kovacs: Stabilizing discrete dynamical systems by monotone Krasnoselskij type iterative schemes, Creat. Math. Inform. 17 (2008) 298–307.Google Scholar
  7. M. G. Cojocaru, P. Daniele, A. Nagurney: Projected dynamical systems, evolutionary variational inequalities, applications, and a computational procedure, in: Pareto Optimality, Game Theory and Equilibria, A. Chinchuluun, A. Migdalas, P. M. Pardalos, L. Pitsoulis (editors), Springer Optimization and its Applications, Springer, New York (2008) 387–406.https://doi.org/10.1007/978-0-387-77247-9_14Google Scholar
  8. A. L. Dontchev, R. T. Rockafellar: Implicit Functions and Solution Mappings: A View from Variational Analysis, Springer, New York (2009).https://doi.org/10.1007/978-0-387-87821-8Google Scholar
  9. P. Dupuis, A. Nagurney: Dynamical systems and variational inequalities, Ann. Oper. Res. 44 (1993) 9–42.https://doi.org/10.1007/BF02073589Google Scholar
  10. K. Goebel, W. A. Kirk: Topics in Metric Fixed Point Theory, Cambridge Studies in Advances Mathematics 28, Cambridge University Press, Cambridge (1990).https://doi.org/10.1017/CBO9780511526152Google Scholar
  11. B. He, X.-Z. He, H. X. Liu, T. Wu: Self-adaptive projection method for co-coercive variational inequalities, Eur. J. Oper. Res. 196 (2009) 43–48.https://doi.org/10.1016/j.ejor.2008.03.004Google Scholar
  12. H. Iiduka: Iterative algorithm for solving triple-hierarchical constrained optimization problem, J. Optim. Theory Appl. 148 (2011) 580–592.https://doi.org/10.1007/s10957-010-9769-zGoogle Scholar
  13. S. Ishikawa: Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147–150.https://doi.org/10.1090/S0002-9939-1974-0336469-5Google Scholar
  14. M. Li, L.-Z. Liao, H.-M. Yuan: A modified descent method for co-coercive variational inequalities, Eur. J. Oper. Res. 189 (2008) 310–323.https://doi.org/10.1016/j.ejor.2007.05.038Google Scholar
  15. W. R. Mann: Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–510.https://doi.org/10.1090/S0002-9939-1953-0054846-3Google Scholar
  16. Z. Opial: Nonexpansive and Monotone Mappings in Banach Spaces, Lecture Notes 67-1, Center for Dynamical Systems, Brown University, Providence (1967).Google Scholar
  17. Z. Opial: Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597.https://doi.org/10.1090/S0002-9904-1967-11761-0Google Scholar
  18. M. Păcurar: Iterative Methods for Fixed Point Approximation, Editura Risoprint, Cluj-Napoca (2009).Google Scholar
  19. W. V. Petryshyn: Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl. 14 (1966) 274–284.https://doi.org/10.1016/0022-247X(66)90027-8Google Scholar
  20. I. A. Rus: An abstract point of view on iterative approximation of fixed points: impact on the theory of fixed point equations, Fixed Point Theory 13 (2012) 179–192.Google Scholar
  21. P. A. Samuelson: Spatial price equilibrium and linear programming, Amer. Econ. Rev. 42 (1952) 283–303.Google Scholar
  22. G. Stampacchia: Formes bilinéaires coercitives sur les ensembles convexes, C. R. Acad. Sci. Paris 258 (1964) 4413–4416.Google Scholar
  23. E. H. Zarantonello (editor): Contributions to Nonlinear Functional Analysis, Proceedings of a Symposium Conducted by the Mathematics Research Center, University of Wisconsin-Madison, April 12-14, 1971, Academic Press, New York (1971).Google Scholar

Suggested citation

E. Toscano, C. Vetro. “Fixed Point Iterative Schemes for Variational Inequality Problems.” Journal of Convex Analysis 25 (2018), No. 2, 701–715. https://doi.org/10.68381/jca25042

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Published by Heldermann Verlag, 2018. Rights now held by Banach Press.