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Studies on Common Solutions of a Variational Inequality and a Fixed-Point Problem

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Abstract

In this paper, we construct a new projection method and prove the strong convergence for finding a common element of a set of fixed points of strict pseudo-contractions and a set of solutions of a variational inequality with inverse strongly monotone mappings in the setting of real Hilbert spaces. Our results improve and extend the recent results by Huang, Noor, and Al-Said (J. Optim. Theory Appl. 147(1):194–204, 2010), Takahashi and Toyoda (J. Optim. Theory Appl. 118(2):417–428, 2003).

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References

  1. Stampacchia, G.: Formes bilineaires coercitives sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 258, 4413–4416 (1964)

    MathSciNet  MATH  Google Scholar 

  2. Bnouhachem, A., Noor, M.A.: A new iterative method for variational inequalities. Appl. Math. Comput. 182, 1673–1682 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Facchinei, F., Pang, J.S.: Finite-Dimensional Variational Inequalities and Complementarity Problems, vol. II. Springer, New York (2003)

    Google Scholar 

  4. Huang, Z.Y., Noor, M.A., Al-Said, E.: On an open question of Takahashi for nonexpansive mappings and inverse strongly monotone mappings. J. Optim. Theory Appl. 147(1), 194–204 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lions, J.L., Stampacchia, G.: Variational inequalities. Commun. Pure Appl. Math. 20, 493–512 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  6. Noor, M.A.: Some developments in general variational inequalities. Appl. Math. Comput. 152, 199–277 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Noor, M.A.: Principles of Variational Inequalities. Lambert Academic Publishing, Saarbrucken (2009)

    Google Scholar 

  8. Noor, M.A.: Extended general variational inequalities. Appl. Math. Lett. 22, 182–185 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Noor, M.A., Rassias, T.M.: Projection methods for monotone variational inequalities. J. Math. Anal. Appl. 237, 405–412 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  10. Solodov, M.V., Svaiter, B.F.: A new projection method for variational inequality problems. SIAM J. Control Optim. 37(3), 765–776 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Takahashi, W., Toyoda, M.: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl. 118(2), 417–428 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Zarantonello, E.H.: Projections on convex sets in Hilbert space and spectral theory. In: Zarantonello, E.H. (ed.) Contributions to Nonlinear Functional Analysis, pp. 237–424. Academic Press, New York (1971)

    Google Scholar 

  13. Zhu, D.L., Marcotte, P.: Coercivity and its role in the convergence of iterative schemes for solving variational inequalities. SIAM J. Optim. 6, 714–726 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

  15. Browder, F.E., Petryshyn, W.V.: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 20(2), 197–228 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tan, K.K., Xu, H.K.: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 178, 301–308 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  17. Liu, Q.H.: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings. Nonlinear Anal. 26, 1835–1842 (1996)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The authors are grateful to the referees and editors for their constructive and valuable suggestions improving this paper.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, P.R. China

    Z. Y. Huang

  2. Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan

    M. A. Noor

Authors
  1. Z. Y. Huang
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  2. M. A. Noor
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Corresponding author

Correspondence to Z. Y. Huang.

Additional information

Communicated by Byung-Soo Lee.

The research of Professor Dr. Z.Y. Huang is supported by National Natural Science Foundation of China (NSFC Grant No. 10871092), supported by the Fundamental Research Funds for the Central University of China (Grant No. 1113020301 and Grant No. 1116020301), and supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD Grant).

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Huang, Z.Y., Noor, M.A. Studies on Common Solutions of a Variational Inequality and a Fixed-Point Problem. J Optim Theory Appl 154, 525–535 (2012). https://doi.org/10.1007/s10957-012-0010-0

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  • DOI: https://doi.org/10.1007/s10957-012-0010-0

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