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An Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces

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Abstract

In this paper, we introduce a new explicit iterative algorithm for finding a solution for variational inequalities over the set of common fixed points of a finite family of nonexpansive maps on Hilbert spaces. An application and a numerical result are given for illustration.

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References

  1. Antman, S.: The influence of elasticity in analysis: modern developments. Bull. Am. Math. Soc. 9(3), 267–291 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Fichera, G.: La nascita della teoria delle diseguaglianze variazionali ricordata dopo trent’anni. Accad. Naz. Lincei. 114, 47–53 (1995)

    Google Scholar 

  3. Stampacchia, G.: Formes bilineares coercitives sur les ensembles convexes. C. R. Hebd. Séances Acad. Sci. 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

  4. Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  5. Duvaut, D., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  6. Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)

    MATH  Google Scholar 

  7. Hlavacek, I., Haslinger, J., Necas, J., Lovicek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1982)

    Google Scholar 

  8. Glowinski, R.: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)

    MATH  Google Scholar 

  9. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)

    Book  MATH  Google Scholar 

  10. Zeidler, E.: Nonlinear Functional Analysis and Its Applications. Springer, New York (1985)

    MATH  Google Scholar 

  11. Aoyama, K., Iiduka, H., Takahashi, W.: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. (2006). doi:10.1155/35390

    MathSciNet  Google Scholar 

  12. Yamada, Y.: The hybrid steepest-descent method for variational inequalities problems over the intersection of the fixed point sets of nonexpansive mappings. In: Butnariu, D., Censor, Y., Reich, S. (eds.) Inhently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473–504. North-Holland, Amsterdam (2001)

    Google Scholar 

  13. Xu, H.K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116, 659–678 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Halpern, B.: Fixed points of nonexpansive mappings. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MATH  Google Scholar 

  15. Xu, H.K., Kim, T.H.: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185–201 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Zeng, L.C., Ansari, Q.H., Wu, S.Y.: Strong convergence theorems of relaxed hybrid steepest-descent methods for variational inequalities. Taiwan. J. Math. 10(1), 13–29 (2006)

    MathSciNet  MATH  Google Scholar 

  17. Zeng, L.C., Wong, N.C., Yao, J.Ch.: Convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities. J. Optim. Theory Appl. 132, 51–69 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, L., Yao, S.S.: Hybrid iteration method for fixed points of nonexpansive mappings. Thai J. Math. 5(2), 183–190 (2007)

    MathSciNet  MATH  Google Scholar 

  19. Mao, Y., Li, J.: Weak and strong convergence of an iterative method for nonexpansive mappings in Hilbert spaces. Appl. Anal. Discrete Math. 2, 197–204 (2008)

    Article  MathSciNet  Google Scholar 

  20. Maingé, P.E.: Extension of the hybrid steepest descent method to a class of variational inequalities and fixed point problems with nonself-mappings. Numer. Funct. Anal. Optim. 29(7–8), 820–834 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Marino, G., Xu, H.K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318, 43–52 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kang, J., Su, Y., Zhang, X.: General iterative for nonexpansive semigroups and variational inequalities in Hilbert spaces. J. Inequal. Appl. (2010). doi:101155/2010/264052

    MathSciNet  Google Scholar 

  23. Mohamedi, I.: Iterative methods for variational inequalities over the intersection of the fixed point set of a semigroup in Banach spaces. Fixed Point Theory Appl. (2011). doi:10.1155/620284

    Google Scholar 

  24. Jung, J.S.: A general iterative scheme for k-strictly pseudo-contractive mappings and optimization problems. Appl. Math. Comput. 217(2), 5581–5588 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, X.: Cui, Y.: The common minimal-norm fixed point of a finite family of nonexpansive mappings. Nonlinear Anal. 73, 76–83 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  26. Marino, G., Xu, H.K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329, 336–346 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Math., vol. 28. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  28. Suzuki, T.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Am. Math. Soc. 135, 99–106 (2007)

    Article  MATH  Google Scholar 

  29. Zhou, H.: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 69, 456–462 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Andrews, H.C., Hunt, B.R.: Digital Image Restoration. Prentice Hall, Englewood Cliffs (1977)

    Google Scholar 

  31. Demoment, G.: Image reconstration and restoration: Overview of common estimation structures and problems. IEEE Trans. Acoust. Speech Signal Process., 37(12), 243–253 (1985)

    MathSciNet  Google Scholar 

  32. Stark, H.: Image Recovery: Theory and Applications. Academic Press, San Diego (1987)

    MATH  Google Scholar 

  33. Combettes, P.L.: Convex set theoretic image recovery by extrapolated iterations of parallel subgradients projections. IEEE Trans. Image Process. 6, 493–506 (1997)

    Article  Google Scholar 

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Correspondence to Nguyen Buong.

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Communicated by Roland Glowinski.

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Buong, N., Duong, L.T. An Explicit Iterative Algorithm for a Class of Variational Inequalities in Hilbert Spaces. J Optim Theory Appl 151, 513–524 (2011). https://doi.org/10.1007/s10957-011-9890-7

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