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An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators

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Abstract

In this paper, we introduce an iterative scheme based on a viscosity approximation method with a modified extragradient method for finding a common solutions of a general system of variational inequalities for two inverse-strongly accretive operator and solutions of fixed point problems involving the nonexpansive mapping in Banach spaces. Consequently, we obtain new strong convergence theorems in the frame work of Banach spaces. Our results extend and improve the recent results of Qin et al. (J Comput Appl Math 233:231–240, 2009) and many others.

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References

  1. Aoyama, K., Iiduka, H., Takahashi, W.: Weak convergence of an iterative sequence for accretive operators in Banach spaces. In: Fixed Point Theory and Applications, vol. 2006, article 35390, pp. 1–13

  2. Browder F.E.: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. 53, 1272–1276 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruck R.E.: Properties of fixed point sets of nonexpansive mappings in Banach spaces. Trans. Am. Math. Soc. 179, 251–262 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ceng L.-C., Wang C.-Y., Yao J.-C.: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 67, 375–390 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cho Y.J., Qin X.: Viscosity approximation methods for a finite family of m-accretive mappings in reflexive Banach spaces. Positivity 12, 483–494 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cho Y.J., Yao Y., Zhou H.: Strong convergence of an iterative algorithm for accretive operators in Banach spaces. J. Comput. Anal. Appl. 10(1), 113–125 (2008)

    MathSciNet  MATH  Google Scholar 

  7. Hao Y.: Iterative algorithms for inverse-strongly accretive mappings with applications. J. Appl. Math. Comput. 31, 193–202 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Katchang P., Khamlae Y., Kumam P.: A viscosity iterative schemes for inverse-strongly accretive operators in Banach spaces. J. Comput. Anal. Appl. 12(3), 678–686 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Kitahara S., Takahashi W.: Image recovery by convex combinations of sunny nonexpansive retractions. Methods Nonlinear Anal. 2, 333–342 (1993)

    MathSciNet  MATH  Google Scholar 

  10. Osilike M.O., Igbokwe D.I.: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 40, 559–567 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  11. Qin X., Cho S.Y., Kang S.M.: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 233, 231–240 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Reich S.: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 44(1), 57–70 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  13. Reich S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Suzuki T.: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305(1), 227–239 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. Xu H.K.: Inequalities in Banach spaces with applications. Nonlinear Anal. 16, 1127–1138 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Xu H.K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298, 279–291 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  17. Yao Y., Noor M.A., Noor K.I., Liou Y.-C., Yaqoob H.: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 110, 1211–1224 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok, 10140, Thailand

    Phayap Katchang & Poom Kumam

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  1. Phayap Katchang
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  2. Poom Kumam
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Correspondence to Poom Kumam.

Additional information

P. Kumam was supported by the Commission on Higher Education and the Thailand Research Fund under Grant MRG5380044.

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Katchang, P., Kumam, P. An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators. Positivity 15, 281–295 (2011). https://doi.org/10.1007/s11117-010-0074-8

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  • DOI: https://doi.org/10.1007/s11117-010-0074-8

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