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Applications of Equilibrium Problems to a Class of Noncoercive Variational Inequalities

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Abstract

In this paper, we are interested in the existence of solutions for a class of noncoercive variational inequalities involving a p-Laplacian type operator. Our approach is based essentially on equilibrium problems and arguments from recession analysis. Our results are of two types: the first is obtained in a monotone framework; the second is obtained without a monotonicity assumption.

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References

  1. 1. Baek, J. S., Properties of Solutions to a Class of Quasilinear Elliptic Problems, PhD Dissertation, Seoul National University, Seoul, Korea, 1992.

  2. 2. Le, V. K., Some Degree Calculations and Applications to Global Bifurcation Results of Variational Inequalities, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 37, pp. 473–500, 1999.

    Article  Google Scholar 

  3. 3. Schuricht, F., Bifurcation from Minimax Solutions by Variational Inequalities in Convex Sets, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 26, pp. 91–112, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  4. 4. Szulkin, A., Existence and Nonuniqueness of Solutions of a Noncoercive Elliptic Variational Inequality, Proceedings of the Symposia on Pure Mathematics, Vol. 45, pp. 413–418, 1986.

    MathSciNet  Google Scholar 

  5. 5. Yang, J. F., Positive Solutions of Quasilinear Elliptic Obstacle Problems with Critical Exponents, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 25, pp. 1283–1306, 1995.

    Article  MATH  Google Scholar 

  6. 6. Yang, J. F., Regularity of Weak Solutions to Quasilinear Elliptic Obstacle Problems, Acta Mathematica Scientia, Vol. 17, pp. 159–166, 1997.

    MATH  Google Scholar 

  7. 7. Zhou, Y., and Huang, Y., Existence of Solutions for a Class of Elliptic Variational Inequalities, Journal of Mathematical Analysis and Applications, Vol. 250, pp. 187–195, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  8. 8. Zhou, Y., and Huang, Y., Several Existence Theorems for the Nonlinear Complementarity Problems, Journal of Mathematical Analysis and Applications, Vol. 202, pp. 776–784, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  9. 9. Hadjisavvas, N., and Schaible, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, pp. 95–111, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  10. 10. Konnov, I. V., and Yao, J. C., On the Generalized Vector Variational Inequality Problem, Journal of Mathematical Analysis and Applications, Vol. 206, pp. 42–58, 1997.

    Article  MATH  MathSciNet  Google Scholar 

  11. 11. Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994.

    MATH  MathSciNet  Google Scholar 

  12. 12. Chadli, O., Chbani, Z., and Riahi, H., Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299–323, 2000.

    Article  MATH  MathSciNet  Google Scholar 

  13. 13. Bianchi, M., and Schaible, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996.

    Article  MATH  MathSciNet  Google Scholar 

  14. 14. Adly, S., Goeleven, D., and Théra, M., Recession Methods in Monotone Variational Hemivariational Inequalities, Journal of Topological Methods in Nonlinear Analysis, Vol. 5, pp. 397–409, 1995.

    MATH  Google Scholar 

  15. 15. Tomarelli, F., Noncoercive Variational Inequalities for Pseudomonotone Operators, Rendiconti del Seminario Matematico e Fisico di Milano, Vol. 61, pp. 141–183, 1991.

    MATH  MathSciNet  Google Scholar 

  16. 16. Chadli, O., Chbani, Z., and Riahi, H., Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discrete and Continuous Dynamical Systems, Vol. 5, pp. 185–195, 1999.

    MATH  MathSciNet  Google Scholar 

  17. 17. Chadli, O., Schaible, S, and Yao, J. C., Regularized Equilibrium Problems and Application to Noncoercive Hemivariational Inequalities, Journal of Optimization Theory and Applications, Vol. 121, pp. 571–596, 2004.

    Article  MATH  MathSciNet  Google Scholar 

  18. 18. Kim, J., and Ku, H., Existence of Solutions for p-Laplacian Type Equations, Journal of the Korean Mathematical Society, Vol. 25, pp. 291–307, 1986.

    MathSciNet  Google Scholar 

  19. 19. Krasnoselskii, M. A., Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon, Elmsford, NY.

  20. 20. Gwinner, J., Stability of Monotone Variational Inequalities with Various Applications, Variational Inequalities and Network Equilibrium Problems, Edited by F. Giannessi and A. Maugeri, Plenum Publishing Company, New York, NY, pp. 123–144, 1995.

    Google Scholar 

  21. 21. Goeleven, D., Noncoercive Variational Problems and Related Results, Research Notes in Mathematics Series, Pitman, London, UK, 1996.

    Google Scholar 

  22. 22. Nagurney, A., Network Economics: A Variational Inequality Approach, Kluwer, Dordrecht, Holland, 1993.

    MATH  Google Scholar 

  23. 23. Border, K. C., Fixed-Point Theorems with Applications to Economics and Game Theory, Cambridge University Press, Cambridge, UK, 1985.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Economics, Faculty of Economics and Social Sciences, Ibn Zohr University, Hay Dakhla, Agadir, Morocco

    O. Chadli (Assistant Professor)

  2. School of Mathematical Sciences and Computing Technology, Central South University, Hunan, China

    Z. Liu (Professor)

  3. Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung, Taiwan

    J. C. Yao (Professor)

Authors
  1. O. Chadli
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  2. Z. Liu
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  3. J. C. Yao
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Corresponding author

Correspondence to J. C. Yao.

Additional information

The first and the third authors were partially supported by the National Science Council of the Republic of China. The second author was partially supported by NSF, Hunan Province, Grant 04JJY20001. The authors express their sincere thanks to two anonymous referees for careful reading and comments leading to the present version of this paper.

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Chadli, O., Liu, Z. & Yao, J.C. Applications of Equilibrium Problems to a Class of Noncoercive Variational Inequalities. J Optim Theory Appl 132, 89–110 (2007). https://doi.org/10.1007/s10957-006-9072-1

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  • DOI: https://doi.org/10.1007/s10957-006-9072-1

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