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Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle

Abstract

Let be a nonlinear mapping from a nonempty closed invex subset of an infinite-dimensional Hilbert space into. Let be proper, invex, and lower semicontinuous on and let be continuously Fréchet-differentiable on with, the gradient of,-strongly monotone, and-Lipschitz continuous on. Suppose that there exist an, and numbers,, such that for all and for all, the set defined by is nonempty, where and is-Lipschitz continuous with the following assumptions. (i), and. (ii) For each fixed, map is sequentially continuous from the weak topology to the weak topology. If, in addition, is continuous from equipped with weak topology to equipped with strong topology, then the sequence generated by the general auxiliary problem principle converges to a solution of the variational inequality problem (VIP): for all.

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References

  1. Argyros IK, Verma RU: On general auxiliary problem principle and nonlinear mixed variational inequalities. Nonlinear Functional Analysis and Applications 2001,6(2):247–256.

    MATH  MathSciNet  Google Scholar 

  2. Argyros IK, Verma RU: Generalized partial relaxed monotonicity and solvability of nonlinear variational inequalities. Panamerican Mathematical Journal 2002,12(3):85–104.

    MathSciNet  Google Scholar 

  3. Cohen G: Auxiliary problem principle and decomposition of optimization problems. Journal of Optimization Theory and Applications 1980,32(3):277–305. 10.1007/BF00934554

    Article  MATH  MathSciNet  Google Scholar 

  4. Cohen G: Auxiliary problem principle extended to variational inequalities. Journal of Optimization Theory and Applications 1988,59(2):325–333.

    MATH  MathSciNet  Google Scholar 

  5. Eckstein J: Nonlinear proximal point algorithms using Bregman functions, with applications to convex programming. Mathematics of Operations Research 1993,18(1):202–226. 10.1287/moor.18.1.202

    Article  MATH  MathSciNet  Google Scholar 

  6. Eckstein J, Bertsekas DP: On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators. Mathematical Programming Series A 1992,55(3):293–318. 10.1007/BF01581204

    Article  MATH  MathSciNet  Google Scholar 

  7. El Farouq N: Pseudomonotone variational inequalities: convergence of the auxiliary problem method. Journal of Optimization Theory and Applications 2001,111(2):305–326. 10.1023/A:1012234817482

    Article  MathSciNet  Google Scholar 

  8. El Farouq N: Pseudomonotone variational inequalities: convergence of proximal methods. Journal of Optimization Theory and Applications 2001,109(2):311–326. 10.1023/A:1017562305308

    Article  MATH  MathSciNet  Google Scholar 

  9. Karamardian S: Complementarity problems over cones with monotone and pseudomonotone maps. Journal of Optimization Theory and Applications 1976,18(4):445–454. 10.1007/BF00932654

    Article  MATH  MathSciNet  Google Scholar 

  10. Karamardian S, Schaible S: Seven kinds of monotone maps. Journal of Optimization Theory and Applications 1990,66(1):37–46. 10.1007/BF00940531

    Article  MATH  MathSciNet  Google Scholar 

  11. Martinet B: Régularisation d'inéquations variationnelles par approximations successives. Revue Française d'Informatique Recherche Opérationnelle 1970, 4: Ser. R-3, 154–158.

    MathSciNet  Google Scholar 

  12. Naniewicz Z, Panagiotopoulos PD: Mathematical theory of hemivariational inequalities and applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 188. Marcel Dekker, New York; 1995:xviii+267.

    Google Scholar 

  13. Rockafellar RT: Augmented Lagrangians and applications of the proximal point algorithm in convex programming. Mathematics of Operations Research 1976,1(2):97–116. 10.1287/moor.1.2.97

    Article  MATH  MathSciNet  Google Scholar 

  14. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM Journal Control Optimization 1976,14(5):877–898. 10.1137/0314056

    Article  MATH  MathSciNet  Google Scholar 

  15. Verma RU: Nonlinear variational and constrained hemivariational inequalities involving relaxed operators. Zeitschrift für Angewandte Mathematik und Mechanik 1997,77(5):387–391. 10.1002/zamm.19970770517

    Article  MATH  Google Scholar 

  16. Verma RU: Approximation-solvability of nonlinear variational inequalities involving partially relaxed monotone (PRM) mappings. Advances in Nonlinear Variational Inequalities 1999,2(2):137–148.

    MATH  MathSciNet  Google Scholar 

  17. Verma RU: A new class of iterative algorithms for approximation-solvability of nonlinear variational inequalities. Computers & Mathematics with Applications 2001,41(3–4):505–512. 10.1016/S0898-1221(00)00292-3

    Article  MATH  MathSciNet  Google Scholar 

  18. Verma RU: General auxiliary problem principle involving multivalued mappings. Nonlinear Functional Analysis and Applications 2003,8(1):105–110.

    MATH  MathSciNet  Google Scholar 

  19. Verma RU: Generalized strongly nonlinear variational inequalities. Revue Roumaine de Mathématiques Pures et Appliquées. Romanian Journal of Pure and Applied Mathematics 2003,48(4):431–434.

    MATH  Google Scholar 

  20. Verma RU: Nonlinear implicit variational inequalities involving partially relaxed pseudomonotone mappings. Computers & Mathematics with Applications 2003,46(10–11):1703–1709. 10.1016/S0898-1221(03)90204-5

    Article  MATH  MathSciNet  Google Scholar 

  21. Verma RU: Partial relaxed monotonicity and general auxiliary problem principle with applications. Applied Mathematics Letters 2003,16(5):791–796. 10.1016/S0893-9659(03)00084-3

    Article  MATH  MathSciNet  Google Scholar 

  22. Verma RU: Partially relaxed cocoercive variational inequalities and auxiliary problem principle. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(2):143–148. 10.1155/S1048953304305010

    Article  MATH  Google Scholar 

  23. Zeidler E: Nonlinear Functional Analysis and Its Applications. II/B. Nonlinear Monotone Operators. Springer, New York; 1990:i–xvi and 469–1202.

    Book  MATH  Google Scholar 

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Correspondence to Ram U. Verma.

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Verma, R.U. Generalized partially relaxed pseudomonotone variational inequalities and general auxiliary problem principle. J Inequal Appl 2006, 90295 (2006). https://doi.org/10.1155/JIA/2006/90295

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