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Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

Abstract

We show that the general variational inequalities are equivalent to the general Wiener-Hopf equations and use this alterative equivalence to suggest and analyze a new iterative method for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality involving multivalued relaxed monotone operators. Our results improve and extend recent ones announced by many others.

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References

  1. 1.

    Stampacchia G: Formes bilinéaires coercitives sur les ensembles convexes. Comptes Rendus de l'Académie des Sciences 1964, 258: 4413–4416.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. Journal of Optimization Theory and Applications 2003,118(2):417–428. 10.1023/A:1025407607560

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Noor MA: General variational inequalities. Applied Mathematics Letters 1988,1(2):119–122. 10.1016/0893-9659(88)90054-7

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Noor MA: Wiener-Hopf equations and variational inequalities. Journal of Optimization Theory and Applications 1993,79(1):197–206. 10.1007/BF00941894

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Noor MA: Projection-proximal methods for general variational inequalities. Journal of Mathematical Analysis and Applications 2006,318(1):53–62. 10.1016/j.jmaa.2005.05.024

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Verma RU: Generalized variational inequalities involving multivalued relaxed monotone operators. Applied Mathematics Letters 1997,10(4):107–109. 10.1016/S0893-9659(97)00068-2

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Naniewicz Z, Panagiotopoulos PD: Mathematical Theory of Hemivariational Inequalities and Applications. Volume 188. Marcel Dekker, New York, NY, USA; 1995:xviii+267.

    Google Scholar 

  8. 8.

    Shi P: Equivalence of variational inequalities with Wiener-Hopf equations. Proceedings of the American Mathematical Society 1991,111(2):339–346. 10.1090/S0002-9939-1991-1037224-3

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Noor MA: Sensitivity analysis for quasi-variational inequalities. Journal of Optimization Theory and Applications 1997,95(2):399–407. 10.1023/A:1022691322968

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Speck F-O: General Wiener-Hopf Factorization Methods, Research Notes in Mathematics. Volume 119. Pitman, Boston, Mass, USA; 1985:vi+157.

    Google Scholar 

  11. 11.

    Reich S: Constructive techniques for accretive and monotone operators. In Proceedings of the 3rd International Conference on Applied Nonlinear Analysis. Academic Press, New York, NY, USA; 1979:335–345.

    Google Scholar 

  12. 12.

    Noor MA: General variational inequalities and nonexpansive mappings. Journal of Mathematical Analysis and Applications 2007,331(2):810–822. 10.1016/j.jmaa.2006.09.039

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Yongfu Su.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Su, Y., Shang, M. & Qin, X. Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings. J Inequal Appl 2007, 064947 (2007). https://doi.org/10.1155/2007/64947

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Keywords

  • Variational Inequality
  • Iterative Method
  • Monotone Mapping
  • Nonexpansive Mapping
  • Monotone Operator
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