Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2017 (2017), Article ID 3941084, 7 pages
https://doi.org/10.1155/2017/3941084
Research Article

A New Nonsmooth Bundle-Type Approach for a Class of Functional Equations in Hilbert Spaces

1School of Mathematics, Liaoning Normal University, Dalian 116029, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China

Correspondence should be addressed to Jie Shen

Received 13 April 2017; Accepted 2 July 2017; Published 8 August 2017

Academic Editor: Hugo Leiva

Copyright © 2017 Jie Shen et al. 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. Deepmala, “Existence theorems for solvability of a functional equation arising in dynamic programming,” International Journal of Mathematics and Mathematical Sciences, vol. 104, no. 3, pp. 273–244, 2014. View at Google Scholar · View at MathSciNet
  2. Z. Liu and J. S. Ume, “On properties of solutions for a class of functional equations arising in dynamic programming,” Journal of Optimization Theory and Applications, vol. 117, no. 3, pp. 533–551, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  3. Z. Liu, R. P. Agarwal, and S. M. Kang, “On solvability of functional equations and system of functional equations arising in dynamic programming,” Journal of Mathematical Analysis and Applications, vol. 297, no. 1, pp. 111–130, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  4. Z. Liu, Y. Xu, J. S. Ume, and S. M. Kang, “Solutions to two functional equations arising in dynamic programming,” Journal of Computational and Applied Mathematics, vol. 192, no. 2, pp. 251–269, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. Shen and L. P. Pang, “A bundle-type auxiliary problem method for solving generalized variational-like inequalities,” Computers and Mathematics with Applications, vol. 55, pp. 2993–2998, 2008. View at Google Scholar
  6. J. Shen and L.-P. Pang, “An approximate bundle-type auxiliary problem method for solving generalized variational inequalities,” Mathematical and Computer Modelling, vol. 48, no. 5-6, pp. 769–775, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. J. Zhang, Y.-Q. Zhang, and L.-W. Zhang, “A sample average approximation regularization method for a stochastic mathematical program with general vertical complementarity constraints,” Journal of Computational and Applied Mathematics, vol. 280, pp. 202–216, 2015. View at Publisher · View at Google Scholar · View at MathSciNet
  8. G. Salmon, V. H. Nguyen, and J. J. Strodiot, “Coupling the auxiliary problem principle and epiconvergence theory to solve general variational inequalities,” Journal of Optimization Theory and Applications, vol. 104, no. 3, pp. 629–657, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. G. Salmon, J. J. Strodiot, and V. H. Nguyen, “A Perturbed auxiliary problem method for paramonotone multivalued mappings,” in Advances in Convex Analysis and Global Optimization, N. Hadjisavvas and P. P. Ardalos, Eds., vol. 54 of Nonconvex Optimization and Its Applications, pp. 515–529, Kluwer, Dordrecht, Netherlands, 2001. View at Publisher · View at Google Scholar
  10. Y. Sonntag, Convergence au sens de Mosco: thdorie et applications a lapproximation des solutions d’ inequations [Ph.D. thesis], Universite de Porvence, 1982.
  11. C. Lemaréchal, “An extension of Davidon methods to nondifferentiable problems,” Mathematical Programming Study, vol. 3, pp. 95–109, 1975. View at Google Scholar
  12. J. Frédéric Bonnans, J. Charles Gilbert, C. Lemaréchal, and C. Sagastizábal, Numerical Optimization, Springer-Verlag, Berlin Heidelberg, 1997.