About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 612819, 7 pages
http://dx.doi.org/10.1155/2013/612819
Research Article

Existence Theorems for Quasivariational Inequality Problem on Proximally Smooth Sets

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 1 November 2012; Accepted 23 December 2012

Academic Editor: Pavel Kurasov

Copyright © 2013 Jittiporn Suwannawit and Narin Petrot. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Bensoussan, M. Goursat, and J.-L. Lions, “Contrôle impulsionnel et inéquations quasi-variationnelles stationnaires,” vol. 276, pp. A1279–A1284, 1973. View at Zentralblatt MATH · View at MathSciNet
  2. G. Stampacchia, “Formes bilinéaires coercitives sur les ensembles convexes,” vol. 258, pp. 4413–4416, 1964. View at Zentralblatt MATH · View at MathSciNet
  3. M. Alimohammady, J. Balooee, Y. J. Cho, and M. Roohi, “Iterative algorithms for a new class of extended general nonconvex set-valued variational inequalities,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 12, pp. 3907–3923, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. J. Balooee and Y. J. Cho, “Algorithms for solutions of extended general mixed variational inequalities and fixed points,” Optimization Letters, p. 27, 2012. View at Publisher · View at Google Scholar
  5. S. S. Chang, Variational Inequality and Complementarity Problem Theory with Applications, Shanghai Scientific and Technology Literature, Shanghai, China, 1991.
  6. W. Chantarangsi, C. Jaiboon, and P. Kumam, “A viscosity hybrid steepest descent method for generalized mixed equilibrium problems and variational inequalities for relaxed cocoercive mapping in Hilbert spaces,” Abstract and Applied Analysis, vol. 2010, Article ID 390972, 39 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. S.-S. Chang, B. S. Lee, X. Wu, Y. J. Cho, and G. M. Lee, “On the generalized quasi-variational inequality problems,” Journal of Mathematical Analysis and Applications, vol. 203, no. 3, pp. 686–711, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. Y. J. Cho and X. Qin, “Systems of generalized nonlinear variational inequalities and its projection methods,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 12, pp. 4443–4451, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. G. Gu, S. Wang, and Y. J. Cho, “Strong convergence algorithms for hierarchical fixed points problems and variational inequalities,” Journal of Applied Mathematics, vol. 2011, Article ID 164978, 17 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. N.-J. Huang, M.-R. Bai, Y. J. Cho, and S. M. Kang, “Generalized nonlinear mixed quasi-variational inequalities,” Computers & Mathematics with Applications, vol. 40, no. 2-3, pp. 205–215, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. N. J. Huang, “On the generalized implicit quasivariational inequalities,” Journal of Mathematical Analysis and Applications, vol. 216, no. 1, pp. 197–210, 1997. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Jaiboon and P. Kumam, “A general iterative method for solving equilibrium problems, variational inequality problems and fixed point problems of an infinite family of nonexpansive mappings,” Journal of Applied Mathematics and Computing, vol. 34, no. 1-2, pp. 407–439, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. P. Kumam, “Strong convergence theorems by an extragradient method for solving variational inequalities and equilibrium problems in a Hilbert space,” Turkish Journal of Mathematics, vol. 33, no. 1, pp. 85–98, 2009. View at Zentralblatt MATH · View at MathSciNet
  14. X. Qin, M. Shang, and Y. Su, “Strong convergence of a general iterative algorithm for equilibrium problems and variational inequality problems,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1033–1046, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. X. Qin, M. Shang, and H. Zhou, “Strong convergence of a general iterative method for variational inequality problems and fixed point problems in Hilbert spaces,” Applied Mathematics and Computation, vol. 200, no. 1, pp. 242–253, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Suantai and N. Petrot, “Existence and stability of iterative algorithms for the system of nonlinear quasi-mixed equilibrium problems,” Applied Mathematics Letters, vol. 24, no. 3, pp. 308–313, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. H. Yao, Y. C. Liou, and J. C. Yao, “An extra gradient method for fixed point problems and variational inequality problems,” Journal of Inequalities and Applications, vol. 2007, Article ID 38752, 12 pages, 2007. View at Publisher · View at Google Scholar
  18. Y. Yao, Y.-C. Liou, and J.-C. Yao, “A new hybrid iterative algorithm for fixed-point problems, variational inequality problems, and mixed equilibrium problems,” Fixed Point Theory and Applications, vol. 2008, Article ID 417089, 15 pages, 2008. View at Zentralblatt MATH · View at MathSciNet
  19. M. Bounkhel, L. Tadj, and A. Hamdi, “Iterative schemes to solve nonconvex variational problems,” Journal of Inequalities in Pure and Applied Mathematics, vol. 4, no. 1, 14 pages, 2003. View at Zentralblatt MATH · View at MathSciNet
  20. F. H. Clarke, Yu. S. Ledyaev, R. J. Stern, and P. R. Wolenski, Nonsmooth Analysis and Control Theory, vol. 178, Springer, New York, NY, USA, 1998. View at MathSciNet
  21. F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley-Interscience, New York, NY, USA, 1983. View at MathSciNet
  22. F. H. Clarke, R. J. Stern, and P. R. Wolenski, “Proximal smoothness and the lower C2 property,” Journal of Convex Analysis, vol. 2, no. 1-2, pp. 117–144, 1995. View at MathSciNet
  23. Y. J. Cho, J. K. Kim, and R. U. Verma, “A class of nonlinear variational inequalities involving partially relaxed monotone mappings and general auxiliary problem principle,” Dynamic Systems and Applications, vol. 11, no. 3, pp. 333–338, 2002. View at Zentralblatt MATH · View at MathSciNet
  24. A. Moudafi, “An algorithmic approach to prox-regular variational inequalities,” Applied Mathematics and Computation, vol. 155, no. 3, pp. 845–852, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. A. Noor, “Iterative schemes for nonconvex variational inequalities,” Journal of Optimization Theory and Applications, vol. 121, no. 2, pp. 385–395, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. A. Noor, N. Petrot, and J. Suwannawit, “Existence theorems for multivalued variational inequality problems on uniformly prox-regular sets,” Optimization Letters, 2012. View at Publisher · View at Google Scholar
  27. N. Petrot, “Some existence theorems for nonconvex variational inequalities problems,” Abstract and Applied Analysis, vol. 2010, Article ID 472760, 9 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L.-P. Pang, J. Shen, and H.-S. Song, “A modified predictor-corrector algorithm for solving nonconvex generalized variational inequality,” Computers & Mathematics with Applications, vol. 54, no. 3, pp. 319–325, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. R. A. Poliquin, R. T. Rockafellar, and L. Thibault, “Local differentiability of distance functions,” Transactions of the American Mathematical Society, vol. 352, no. 11, pp. 5231–5249, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet