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Strong convergence theorems for fixed point problems for strict pseudo-contractions and variational inequalities for inverse-strongly accretive mappings in uniformly smooth Banach spaces

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Abstract

In this paper, we first study a simple viscosity iterative algorithm for finding a fixed point of a nonexpansive mapping in uniformly smooth Banach space and obtain a strong convergence theorem under suitable conditions. Then, we apply our iterative algorithm to find a common element of the set of fixed points for a strict pseudo-contraction and the set of fixed points for a nonexansive mapping in uniformly smooth Banach space. We also apply our main results to find a common element of the set of fixed points for strict pseudo-contraction and nonexpansive mapping, the set of solutions of general variational inequalities for two inverse-strongly accretive mappings and equilibrium problems in uniformly smooth Banach spaces or Hilbert spaces. Finally, two numerical examples are given in support of our main results.

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References

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Ceng, L.C., Wang, C., 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  Google Scholar 

  4. 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  Google Scholar 

  5. Yao, Y., Maruster, S.: Strong convergence of an iterative algorithm for variational inequalities in Banach spaces. Math. Comput. Model. 54, 325–329 (2011)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  7. Yao, Y., Liou, Y.C., Kang, S.M., Yu, Y.: Algorithms with strong convergence for a system of nonlinear variational inequalities in Banach spaces. Nonlinear Anal. 74, 6024–6034 (2011)

    Article  MathSciNet  Google Scholar 

  8. 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  Google Scholar 

  9. Qin, X., Chang, S.S., Cho, Y.J.: Iterative methods for generalized equilibrium problems and fixed point problems with applications. Nonlinear Anal. Real World Appl. 11, 2963–2972 (2010)

    Article  MathSciNet  Google Scholar 

  10. Sunthrayuth, P., Kumam, P.: Viscosity approximation methods base on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces. Math. Comput. Model. 58, 1814–1828 (2013)

    Article  Google Scholar 

  11. Cai, G., Bu, S.: Convergence analysis for variational inequality problems and fixed point problems in 2-uniformly smooth and uniformly convex Banach spaces. Math. Comput. Model. 55, 538–546 (2012)

    Article  MathSciNet  Google Scholar 

  12. Colao, V., Marino, G., Muglia, L.: On some auxiliary mappings generated by nonexpansive and strictly pseudo-contractive mappings. Appl. Math. Comput. 218, 6232–6241 (2012)

    MathSciNet  MATH  Google Scholar 

  13. Dong, Q.L., He, S., Su, F.: Strong convergence of an iterative algorithm for an infinite family of strict pseudo-contractions in Banach spaces. Appl. Math. Comput. 216, 959–969 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Zhou, H.: Convergence theorems for \(\lambda \)-strict pseudo-contractions in \(2\)-uniformly smooth Banach spaces. Nonlinear Anal. 69, 3160–3173 (2008)

    Article  MathSciNet  Google Scholar 

  15. Zhang, H., Su, Y.: Strong convergence theorems for strict pseudo-contractions in \(q\)-uniformly smooth Banach spaces. Nonlinear Anal. 70, 3236–3242 (2009)

    Article  MathSciNet  Google Scholar 

  16. Song, Y., Chen, R.: Convergence theorems of iterative algorithms for continuous pseudocontractive mappings. Nonlinear Anal. 67, 486–497 (2007)

    Article  MathSciNet  Google Scholar 

  17. Sahu, D.R., Petrusel, A.: Strong convergence of iterative methods by strictly pseudocontractive mappings in Banach spaces. Nonlinear Anal. 74, 6012–6023 (2011)

    Article  MathSciNet  Google Scholar 

  18. 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  Google Scholar 

  19. Cholamjiak, P., Suantai, S.: Strong convergence for a countable family of strict pseudocontractions in q-uniformly smooth Banach spaces. Comput. Math. Appl. 62, 787–796 (2011)

    Article  MathSciNet  Google Scholar 

  20. Shehu, Y.: An iterative approximation of fixed points of strictly pseudocontractive mappings in Banach spaces. Mat. Vesn. 67, 79–91 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Suzuki, T.: Strong convergence of Krasnoselskii and Manns type sequences for one parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 305, 227–239 (2005)

    Article  MathSciNet  Google Scholar 

  22. Liu, L.S.: Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 194, 114–125 (1995)

    Article  MathSciNet  Google Scholar 

  23. Cholamjiak, P., Suantai, S.: Weak convergence theorems for a countable family of strict pseudocontractions in Banach spaces. Fixed Point Theory Appl. 2010, Article ID 632137, 17 (2010)

  24. Song, Y., Chen, R., Zhou, H.: Viscosity approximation methods for nonexpansive mapping sequences in Banach spaces. Nonlinear Anal. 66, 1016–1024 (2007)

    Article  MathSciNet  Google Scholar 

  25. Chancelier, J.P.: Iterative schemes for computing fixed points of nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 353, 141–153 (2009)

    Article  MathSciNet  Google Scholar 

  26. Blum, E., Oettli, W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)

    MathSciNet  MATH  Google Scholar 

  27. Combettes, P.L., Hirstoaga, S.A.: Equilibrium programming in Hilbert space. J. Nonlinear Convex Anal. 6, 117–136 (2005)

    MathSciNet  MATH  Google Scholar 

  28. Cho, S.Y., Kang, S.M.: Approximation of common solutions of variational inequalities via strict pseudocontractions. Acta Math. Sci. 32, 1607–1618 (2012)

    Article  MathSciNet  Google Scholar 

  29. Kopecká, K., Reich, S.: Nonexpansive retracts in Banach spaces. Banach Cent. Publ. 77, 161–174 (2007)

    Article  MathSciNet  Google Scholar 

  30. Censor, Y., Gibali, A., Reich, S.: Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim. Methods Softw. 26, 827–85 (2011)

    Article  MathSciNet  Google Scholar 

  31. Gibali, A., Reich, S., Zalas, R.: Outer approximation methods for solving variational inequalities in Hilbert space. Optimization 66, 417–437 (2017)

    Article  MathSciNet  Google Scholar 

  32. Reich, S.: Constructive techniques for accretive and monotone operators. In: Applied Nonlinear Analysis. pp. 335–345. Academic Press, New York (1979)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Gang Cai

  2. Department of Mathematics, University of Nigeria, Nsukka, Nigeria

    Yekini Shehu

  3. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Wisconsin, USA

    Olaniyi Samuel Iyiola

Authors
  1. Gang Cai
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  2. Yekini Shehu
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  3. Olaniyi Samuel Iyiola
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Correspondence to Gang Cai.

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This work was supported by the NSF of China (Grant nos. 11401063, 11771063), the Natural Science Foundation of Chongqing (Grant nos. cstc2017jcyjAX0006, cstc2016jcyjA0116), Science and Technology Project of Chongqing Education Committee (Grant nos. KJ1703041, KJZDM201800501, KJ16003162016), the University Young Core Teacher Foundation of Chongqing (Grant nos. 020603011714), Talent Project of Chongqing Normal University (Grant no. 02030307-00024).

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Cai, G., Shehu, Y. & Iyiola, O.S. Strong convergence theorems for fixed point problems for strict pseudo-contractions and variational inequalities for inverse-strongly accretive mappings in uniformly smooth Banach spaces. J. Fixed Point Theory Appl. 21, 41 (2019). https://doi.org/10.1007/s11784-019-0677-z

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  • DOI: https://doi.org/10.1007/s11784-019-0677-z

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