Approximation Theory and its Applications

, Volume 11, Issue 3, pp 95–105 | Cite as

On interative algorithms for a class of nonlinear variational inequalities

  • M. A. Noor


In this paper we use the auxiliary principle technique to suggest and analyze novel and innovative iterative algorithms for a class of nonlinear variational inequalities. Several special cases, which can be obtained from our main results, are also discussed.


Variational Inequality Complementarity Problem Variational Inequality Problem Auxiliary Problem General Variational Inequality 
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  1. [1]
    Baiocchi, G. and Capelo, A., Variational and Quasi-variational Inequalities, J. Wiley and Sons, New York, London, 1984.Google Scholar
  2. [2]
    Cohen, G., Auxiliary Problem Principle Extended to Variational Inequalities, J. Optim. Theory Appl., 59(1988), 325–333.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    Crank, J., Free and Moving Boundary Problems, Clarendon Press, Oxford, U. K., 1984.MATHGoogle Scholar
  4. [4]
    Duvaut, G. and Lions, J. L., Inequalities in Mechanics and Physics, Springer-Verlag, Berlin, 1976.MATHGoogle Scholar
  5. [5]
    Fichera, G., Problemi Elastostatici con Vincoli Unilaterali: il Problema di Signorini con Ambigue Condizioni al Contorno, Atti Acad. Naz. Lincei. Mem. Cl. Sci. Fiz. Mat. Nat. Sez. Ia. 7(8) (1963–64), 91–140.MathSciNetGoogle Scholar
  6. [6]
    Filippov, V. M., Variational Principles for Nonpotential Operators, Amer. Math. Soc. Trans. Math. Monographs, Vol. 77,1989.Google Scholar
  7. [7]
    Fukushima, M., Equivalent Differentiable Optimization Problems and Descent Methods for Asymmetric Variational Inequality Problem, Mathematical Programming, 53 (1992), 99–110.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Glowinski, R., Lions, J. L. and Tremolieres, R., Numerical Analysis of Variational Inequalities, North-Holland, Amsterdam, 1981.MATHGoogle Scholar
  9. [9]
    Kikuchi, N. and Σden, J. T., Contact Problems in Elasticity, SIAM Publishing Co. Philadelphia, U. S. A., 1988.MATHGoogle Scholar
  10. [10]
    Larsson, T. and Patriksson, M., A Class of Gap Functions for Variational Inequalities, Mathematical Programming, 64(1994),53–79.CrossRefMathSciNetGoogle Scholar
  11. [11]
    Lions, J. L. and Stampacchia, G., Variational Inequalities, Comm. Pure Appl. Math. 20(1967), 493–519.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    Noor, M. Aslam, The Riesz-Fréchet Theorem and Monotonicity, M. Sc. Thesis, Queen's University, Kingston, Canada, 1971.Google Scholar
  13. [13]
    Noor, M. Aslam, Equivalence of Differentiable Optimization Problems for Variational Inequalities, J. Nat. Geometry (1995) to appear.Google Scholar
  14. [14]
    ——, Variational Inequalities in Physical Oceanography, in: Ocean Waves Engineering, 201–226. M. Rahman ed., Computational Mechanics Publications, Southampton, U. K. (1994), 201–226.Google Scholar
  15. [15]
    Noor, M. Aslam, Noor, K. Inayat and Rassias, Th. M., Some Aspects of Variational Inequalities, J. Comput. Appl. Math. 47(1993),285–312.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    Noor, M. Aslam, General Variational Inequalities, Appl. Math. Letters 1(1988),119–122.MATHCrossRefGoogle Scholar
  17. [17]
    ——, General Algorithm and Sensitivity Analysis for Variational Inequalities, J. Appl. Math. Stoch. Anal. 5(1992), 29–42.MATHCrossRefGoogle Scholar
  18. [18]
    ——, Wiener-Hopf Equations and Variational Inequalities, J. Optim. Theory Appl., 79(1993), 197–206.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    ——, General Nonlinear Variational Inequalities, J. Math. Anal. Appl., 126(1987),78–84.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    ——, General Algorithm for Variational Inequalities, J. Optim. Theory Appl., 73(1992), 409–413.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    ——, Mixed Variational Inequalities, Appl. Math. Lett. 3(1990), 73–75.MATHCrossRefGoogle Scholar
  22. [22]
    ——, General Algorithm for Variational Inequalities (I), Math. Japonica, 38(1993), 47–53MATHGoogle Scholar
  23. [23]
    ——, General quasi Complementarity problem, Math. Japonica, 36(1991),113–119.MATHGoogle Scholar
  24. [24]
    Oden, J. T. and Kikuchi, N., Theory of Variational Inequalities with Application to Fluid Flow Through Porous Media, Int. J. Engng. Sci, 18(1980),1173–1284.MATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    Oettli, W., Some Remarks on General Complementarity Problems and Quasi-Variational Inequalities, Unviersity of Mannheim, Germany, Pre-print, 1987.Google Scholar
  26. [26]
    Petryshyn, W. V., Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space, Trans. Amer. Math. Soc., 105(1962),136–175.MATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    Shi, P., An Iterative Method for Obstacle Problems via Green's Functions, Nonlin. Anal. Theory Methods Appl., 15(1990),339–344.MATHCrossRefGoogle Scholar
  28. [28]
    Smith, M. J., The Existence, Uniqueness and Stability of Traffic Equilibra, Trans. Science, 14 (1980),42–54.CrossRefGoogle Scholar
  29. [29]
    Speck, F. O., General Wiener-Hopf Factorization Methods, Research Notes in Mathematics 119, Pitman Advanced Publishing Program, 1985.Google Scholar
  30. [30]
    Stampacchia, G., Formes Bilinearires Coercitives sur les Ensembles Convexes, C. R. Acad. Sci. Paris, 258(1964),4413–4416.MATHMathSciNetGoogle Scholar
  31. [31]
    Taji, K., Fukushima, M. and Ibaraki, T., A Globally Convergent Newton Method for Solving Monotone Variational Inequalities, Mathematical Programming, 58(1993),369–383.CrossRefMathSciNetGoogle Scholar
  32. [32]
    Wu, J. H., Florian, M. and Marcotte, P., A General Descent Framework forthe Monotone Variational Inequality Problem, Mathematical Programming, 61(1993),281–300.CrossRefMathSciNetGoogle Scholar
  33. [33]
    Zhu, D. L. and Marcotte, P., An Extended Descent Framework for Variational Inequalities, J. Optim. Theory Applicaitons, 80(1994),349–366.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 1995

Authors and Affiliations

  • M. A. Noor
    • 1
  1. 1.Department of Mathematics College of ScienceKing Saud UniversityRiyadhSaudi Arabia

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